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Approximate Set Covering in Uniform Hypergraphs
JOURNAL OF ALGORITHMS ARTICLE NO.
25, 118]143 Ž1997.
AL970872
Approximate Set Covering in Uniform Hypergraphs Michael KrivelevichU Department of Mathematics, Raymond and Be¨ erly Sackler Faculty of Exact Sciences, Tel-A¨ i¨ Uni¨ ersity, Tel-A¨ i¨ , Israel Received November 29, 1995; revised April 23, 1997
The weighted set covering problem, restricted to the class of r-uniform hypergraphs, is considered. We propose a new approach, based on a recent result of Aharoni, Holzman, and Krivelevich about the ratio of integer and fractional covering numbers in k-colorable r-uniform hypergraphs. This approach, applied to hypergraphs of maximal degree bounded by D, yields an algorithm with approximation ratio r Ž1 y crD1r Ž ry1. .. Next, we combine this approach with an adaptation of the local ratio theorem of Bar-Yehuda and Even for hypergraphs and present a general framework of approximation algorithms, based on subhypergraph exclusion. An application of this scheme is described, providing an algorithm with approximation ratio r Ž1 y crnŽ ry1.r r . for hypergraphs on n vertices. We discuss also the limitations of this approach. Q 1997 Academic Press
1. INTRODUCTION The minimum set co¨ er problem is certainly one of the most central problems of combinatorial optimization. In an instance of this problem, a collection C of subsets of a finite set S and a weight function w: C ª Rq are given, and the task is to find a subset CX : C of minimum weight w Ž CX . s Ý c g C X w Ž c . such that every element of S belongs to at least one member of CX . When all weights equal identically 1, we obtain the unweighted version of the problem. For our purposes it is more convenient to reformulate this problem in terms of hypergraphs. A hypergraph H is an ordered pair H s Ž V, E ., where V is a finite nonempty set Žthe set of ¨ ertices. and E is a collection of distinct nonempty subsets of V Žthe set of edges.. H is called r-uniform * This research forms part of a Ph.D. thesis written by the author under the supervision of Professor Noga Alon. Research supported in part by a Charles Clore Fellowship. 118 0196-6774r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
APPROXIMATE SET COVERING
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if < e < s r for all edges e g EŽ H .. The set cover problem can be represented as follows: for given hypergraph H s Ž V, E . and weight function w: V ª Rq, solve the optimization problem min
Ý wŽ ¨ . g Ž ¨ .
¨ gV
s.t.
Ý gŽ¨. G 1
for every e g E Ž H . ,
¨ ge
g Ž ¨ . g 0, 1 4
for every ¨ g V .
Ž 1.
This formulation leads naturally to the fractional relaxation of the integer problem Ž1.: min
Ý wŽ ¨ . g Ž ¨ .
¨ gV
s.t.
Ý gŽ¨. G 1
for every e g E Ž H . ,
¨ ge
g Ž ¨ . G 0 for every ¨ g V .
Ž 2.
For a hypergraph H s Ž V, E ., any feasible solution g of Ž1. Žalternatively, the set of vertices of weight 1 in a feasible solution g . is called a co¨ er of H with ¨ alue < g < s ݨ g V w Ž ¨ . g Ž ¨ .. An optimal solution of Ž1. is an optimal co¨ er of H, having the co¨ ering number of H wwhich we denote by t Ž H .x as its value. Quite similarly, any feasible solution g of Ž2. is a fractional co¨ er of H; an optimal solution of Ž2. is an optimal fractional co¨ er of H with value t U Ž H ., called the fractional co¨ ering number of H. It is important to note that the problem Ž2. is a linear programming problem that can be solved in time polynomial in < V < and < E < w17x. Clearly, the solution t U Ž H . of the fractional problem Ž2. may serve as a lower bound on the solution t Ž H . of the more difficult integer problem Ž1.. In this paper we consider the case of r-uniform hypergraphs, where r is thought to be a fixed number. The above-defined problem Ž1. turns out to be NP-complete even for the unweighted case and r s 2 Žthat is, for the case of graphs, where this problem is usually called the ¨ ertex co¨ er problem., as shown by Karp w18x. This result motivates studying approximate algorithms for this problem. An approximation algorithm A for problem Ž1. is a polynomial time algorithm which, for a given input instance Ž H, w . of Ž1., produces a feasible solution. The value of the solution obtained by A for an instance H is denoted by AŽ H .. The approximation ratio R A of an approximation algorithm A on a family of instances H is R A s sup AŽ H .rt Ž H .: H g H 4 .
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MICHAEL KRIVELEVICH
The approximation ratio R A can be viewed as a quantitative measure of the quality of an algorithm A. Here we aim to develop efficient approximation algorithms for the class Hr of r-uniform hypergraphs for a given and fixed value of r. The set cover problem remains hard even as an approximation problem. Speaking explicitly, it is MAX-SNP complete as proven in w22x. Moreover, a commonly believed conjecture Žsee, e.g., w15, 23x. states that unless P s NP, there does not exist a polynomial time approximation algorithm A with approximation ratio R A F r y e for any positive constant e . Let us briefly describe known results in the positive direction. We start with the first nontrivial case r s 2. The following very simple algorithm, attributed in w10x to Gavril, can be seen easily to have approximation ratio 2 in the unweighted case. The algorithm proceeds as follows. Given a graph G, find a maximal Žunder inclusion. matching M in G Žfor example, by picking edges greedily. and take all vertices in the edges of M as a cover. This idea can be easily generalized to the case of general r. Several algorithms having approximation ratio 2 for the general Žweighted. case are known; the algorithm of Bar-Yehuda and Even w3x is the simplest of them. Most Žif not all. of the more sophisticated approximation algorithms rely heavily on the theorem of Nemhauser and Trotter w21x, which enables reduction of a given problem to the problem restricted to a class of graphs G satisfying < V Ž G .
APPROXIMATE SET COVERING
121
so-called local ratio theorem and proceeds by excluding subgraphs G 0 : G that have large covering number relative to their order. Returning to the set covering problem for r-uniform hypergraphs for general r, we mention the algorithm of Hall and Hochbaum w11x for a general multicovering problem wwhere in Ž1. the first group of restrictions is ݨ g e g Ž ¨ . G bŽ e . for each edge e g E x. This algorithm has approximation ratio r and performs O Žmax< V <, < E <4< V <. iterations. Peleg, Schechtman, and Wool w23x presented several algorithms that have approximation ratio r for the unweighted case. They also described a randomized approximation algorithm that has approximation ratio r Ž1 y Ž crm.1r r ., where m is the number of edges in the hypergraph H and c is some absolute constant. This algorithm first solves the fractional problem Ž2. and then uses the technique of randomized rounding with scaling that was developed by Raghavan and Thompson w24x. Now we describe new results presented in this paper and its structure. In Section 2 we gather some results about fractional covers and matchings that will be used in subsequent sections. A recent theorem of Aharoni, Holzman, and Krivelevich w1x that bounds the ratio between integer and fractional covering numbers in k-colorable r-uniform hypergraphs is presented together with two different proofs in Section 3. Both proofs are algorithmic; the corresponding algorithms are described in the same section. In Section 4 we apply one of these algorithms to the problem of approximating the set covering number for r-uniform hypergraphs of maximal vertex degree bounded by D and obtain an algorithm with approximation ratio r Ž1 y crD1rŽ ry1. .. In Section 5 we combine the second algorithm of Section 3 with the local ratio approach of Bar-Yehuda and Even to give a general scheme of an approximation algorithm parameterized by a family H0 of excluded subhypergraphs. A particular realization for a concrete choice of H0 is presented in Section 6, which yields an algorithm with approximation ratio r Ž1 y crnŽ ry1.r r . for hypergraphs on n vertices. We discuss also the limitations of the local ratio approach in Section 7. We close this section with some notation. Given a hypergraph H s Ž V, E . and a weight function w: V ª Rq, the degree dŽ ¨ . of a vertex ¨ g V is the number of edges of H containing ¨ . For a subset V0 : V, the notation H w V0 x stands for the subhypergraph of H induced by V0 . This subhypergraph has set of edges EŽ V0 .. We denote its cardinality by eŽ V0 . s < EŽ V0 .<. A subset V0 : V is called independent in H if eŽ V0 . s 0. The weight w Ž V0 . of V0 is w Ž V0 . s ݨ g V 0 w Ž ¨ .. For a given hypergraph H0 , a hypergraph H is called H0-free, if H does not contain a Žnot necessarily induced. copy of H0 . Given a family of hypergraphs H0 s H1 , . . . , Ht 4 , H is called H0-free if it is Hi-free for each member Hi of H0 .
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MICHAEL KRIVELEVICH
2. FRACTIONAL COVERS AND MATCHINGS IN HYPERGRAPHS In this short section we accumulate some basic facts about fractional covers and matchings to be used in the sequel. Let us formulate the LP problem dual to the previously defined problem Ž2. of determining the fractional covering number of a hypergraph H s Ž V, E .. max
Ý f Ž e. egE
s.t.
Ý f Ž e. F w Ž ¨ .
for every ¨ g V Ž H . ,
e2¨
f Ž e. G 0
for every e g E.
Ž 3.
A feasible solution f of this problem is a fractional matching in H with
¨ alue < f < s Ý e g E f Ž e .. A fractional matching that has the largest possible
value is called optimal and its value is the fractional matching number n U Ž H .. The duality theorem of linear programming applied to the pair of problems Ž2. and Ž3. implies Proposition 1. PROPOSITION 1. For e¨ ery hypergraph H s Ž V, E . and e¨ ery weight function w: V ª Rq, the following statements hold true: 1. t U Ž H . s n U Ž H .. 2. If g: V ª Rq is an optimal fractional co¨ er and f : E ª Rq is an optimal fractional matching, then f Ž e. ) 0
implies
Ý g Ž ¨ . s 1,
¨ ge
gŽ¨. ) 0
implies
Ý f Ž e. s wŽ ¨ . .
Ž 4.
¨ ge
ŽThese are the so-called complementary slackness conditions.. We will make a direct use of the following proposition. PROPOSITION 2. For e¨ ery r-uniform hypergraph H s Ž V, E . and e¨ ery weight function w: V ª Rq the following statements hold true: 1. If g is a fractional co¨ er of H, then for e¨ ery subset V0 : V the function g X : V0 ª Rq, defined by g X Ž ¨ . s g Ž ¨ . for e¨ ery ¨ g V0 Ž that is, g X is the restriction of g to V0 ., is a fractional co¨ er of the hypergraph H w V0 x. 2. If g is an optimal fractional co¨ er of H, satisfying g Ž ¨ . ) 0 for e¨ ery ¨ g V, then t U Ž H . s w Ž V .rr.
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APPROXIMATE SET COVERING
Proof. 1. Follows immediately from the definition of a fractional cover. 2. Let f : E ª Rq be an optimal fractional matching of H with respect to w. Then, by the complementary slackness conditions Ž4., wŽ V . s
Ý wŽ ¨ . s Ý Ý f Ž e. s Ý f Ž e. < e l V <
¨ gV
s
¨ gV e2¨
egE U
Ý f Ž e . r s r Ý f Ž e . s rn Ž H . s rt U Ž H . . egE
egE
w9x for additional The reader is referred to the survey paper by Furedi ¨ results about integer and fractional covers and matchings in hypergraphs.
3. COVERS IN k-COLORED HYPERGRAPHS Given a hypergraph H s Ž V, E ., a partition V s V1 j ??? j Vk is a proper k-coloring of H with colors V1 , . . . , Vk if < e l Vi < - < e < for each e g EŽ H . with < e < G 2 and 1 F i F k; that is, no edge, besides of course singletons, is monochromatic. A hypergraph H is k-colorable if it has a proper k-coloring. Hochbaum w15x suggested the following approach to vertex cover approximation. Given an instance Ž G, w ., first reduce the problem to an instance Ž G 0 , w ., for which t Ž G 0 . G w Ž V Ž G 0 ..r2 by applying the Nemhauser]Trotter algorithm, then color G 0 by k colors V1 , . . . , Vk , and take the complement of a color class Vi , having the maximal weight w Ž Vi ., to be an approximate solution, thus obtaining the approximation ratio 2 y 2rk. Trying to generalize this idea for the case of r-uniform hypergraphs for general r, we need to bypass the lack of the Nemhauser]Trotter type result for r G 3. This can be achieved by using the following recent result of Aharoni, Holzman, and Krivelevich w1x, whose proof can be converted into a polynomial time approximation algorithm. THEOREM 1. Let H s Ž V, E . be an r-uniform k-colorable hypergraph and let w: V ª Rq be a weight function on the ¨ ertices of H. Denote by t Ž H . and by t U Ž H . the co¨ ering and the fractional co¨ ering numbers of H, respecti¨ ely, with respect to w. Then
tŽH. U
t ŽH.
½
F max r y 1,
ky1 k
5
r .
wWe remark that in w1x bounds on the ratio t Ž H .rt U Ž H . are proven for various types of r-uniform hypergraphs.x For the sake of completeness we provide here a proof of the Theorem 1. Actually, we give two different proofs, to be converted later into two approximation algorithms.
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MICHAEL KRIVELEVICH
Proof 1. Suppose that the theorem fails and let H s Ž V, E . with w: V ª Rq be a counterexample with the smallest number of vertices. Then we must have De g E e s V. Fix some proper k-coloring V1 , . . . , Vk of H. Let g: V ª Rq be an optimal fractional cover of H and let f : E ª Rq be an optimal fractional matching in H with respect to w. We distinguish between two cases. Case 1. has
g Ž ¨ . ) 0 for all ¨ g V. Then, by Proposition 2, part 2, one
tUŽ H . s
wŽ V . r
.
Ž 5.
On the other hand, the complement of a heaviest color class is clearly a cover of H with weight not exceeding ŽŽ k y 1.rk . w Ž V .. Therefore,
tŽH. F
ky1 k
wŽ V . .
Ž 6.
Comparing Ž5. and Ž6., we derive t Ž H .rt U Ž H . F ŽŽ k y 1.rk . r}a contradiction with the choice of H. Case 2. There exists a vertex ¨ 0 g V with g Ž ¨ 0 . s 0. By our assumption about H, this vertex belongs to some edge e0 g E. Whereas < e0 < s r and ݨ g e 0 g Ž ¨ . G 1, there exists a vertex ¨ 1 g e0 with g Ž ¨ 1 . G 1rŽ r y 1.. If ¨ 14 is a cover of H, then clearly t Ž H . F w Ž ¨ 1 .. On the other hand, by the complementary slackness conditions Ž4., we have w Ž ¨ 1 . s Ý e 2 ¨ 1 f Ž e .. Therefore, t U Ž H . s n U Ž H . G Ý e 2 ¨ 1 f Ž e . s w Ž ¨ 1 ., so in fact t Ž H . s t U Ž H . s w Ž ¨ 1 ., contradicting the choice of H. So we may consider the hypergraph H X s H w V y ¨ 1 x. Define a new weight function wX : V Ž H X . ª Rq by wX Ž ¨ . s w Ž ¨ . for all ¨ g V Ž H X .. Whereas H X is also r-uniform and k-colorable, it follows from the minimality of H that the theorem statement is valid for H X . Obviously,
t Ž H . F t Ž H . q wŽ ¨1.
Ž 7.
Ža cover of H X can be extended to a cover of H by adding ¨ 1 .. On the other hand, by Proposition 2, part 1, we conclude that
t U Ž H X . F t U Ž H . y g Ž ¨ 1 . w Ž ¨ 1 . F t U Ž H . y w Ž ¨ 1 . r Ž r y 1. . Ž 8.
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APPROXIMATE SET COVERING
It follows from Ž7. and Ž8. that
t Ž H . F t Ž H X . q w Ž ¨ 1 . F max r y 1,
½
½ ½
F max r y 1, F max r y 1,
ky1 k ky1 k
r
ky1
tU H y
5ž Ž . 5 Ž .
k
r t U Ž HX . q wŽ ¨1.
5
wŽ ¨1. ry1
/
q wŽ ¨1.
r tU H ,
again obtaining a contradiction to the choice of H. Proof 2. We present this proof in the following probabilistic setting. LEMMA 1. Let H s Ž V, E . be an r-uniform hypergraph and let w: V ª Rq be a weight function on the ¨ ertices of H. Let g: V ª Rq be an optimal fractional co¨ er of H with respect to w. Suppose V s V1 j ??? j Vk is a partition of V. Suppose further that for some d ) 0 there exists a set B : w0, d x k such that for e¨ ery x s Ž x 1 , . . . , x k . g B the set k
T Ž x. s
D ¨ g Vi : g Ž ¨ . G x i 4 is1
is a co¨ er of H. If there exists a probability measure m defined on B w m Ž B . s 1x such that all marginal distributions m i , 1 F i F k, are uniform on the inter¨ al w0, d x Ž that is, if x g B is randomly chosen from B according to the measure m , then P w a F x i F b x s Ž b y a.rd for e¨ ery 0 F a F b F d ., then
tŽH. U
t ŽH.
F
1
d
.
Proof. Let x g B be randomly chosen from B according to the measure m. Define a random variable Y s w ŽT Ž x .., where T Ž x . is as defined previously. Let us estimate the expectation of Y. By linearity of expectation, Ew Y x s ݨ g V w Ž ¨ . P w Y¨ s 1x, where Y¨ is the indicator random variable for ¨ g V being selected to T. Whereas m has marginal distributions uniform on the interval w0, d x, for every 1 F i F k and for every ¨ g Vi we have P w Y¨ s 1 x s P w ¨ g T x s P g Ž ¨ . G x i s min 1, g Ž ¨ . rd 4 F g Ž ¨ . rd ; hence, Ew Y x s
Ý w Ž ¨ . P w Y¨ s 1x F Ý w Ž ¨ .
¨ gV
¨ gV
gŽ¨.
d
s
tUŽ H . d
.
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MICHAEL KRIVELEVICH
We conclude that there exists at least one point x g B, for which the corresponding cover T Ž x . has weight w ŽT ., satisfying w ŽT . F t U Ž H .rd . Now our strategy is to find, for a k-colored hypergraph H with a given proper k-coloring V1 , . . . , Vk , an appropriate set B and a measure m and then to apply Lemma 1. Consider first the case k - r. In this case we need to prove t Ž H .rt U Ž H . F r y 1. Whereas every k-colorable hypergraph for k F r is also r-colorable, we may assume without loss of generality that H is an r-colorable hypergraph, thus reducing the second case k G r. Assume now that k G r. Define first k q 1 points Q0 , Q1 , . . . , Q k in w0, krŽŽ k y 1. r .x k as Q0 s
ž
ž
1 r
,...,
Q j s 0, . . . ,
1 r
/
, k
Ž k y 1. r
/
,...,0 ,
^ ` _
j s 1, . . . , k.
jth coord.
Now let Bj be the interval in R k joining Q0 and Q j , and let k
Bs
D Bj . js1
Clearly, B : w0, krŽŽ k y 1. r .x k . To check that T Ž x . is a cover for every x g B, consider, for example, the interval B1 s w Q0 Q1 x. It can be seen easily that for each point x s Ž x 1 , . . . , x k . g B1 we have x 1 G 1rr and x 1 G x i for each 2 F i F k, and the last k y 1 coordinates have the same value which we denote by y. The equation Ž k y 1. x 1 q y s krr is satisfied by both endpoints of B1 and hence by every point of B1. Consider an edge e g E and denote < e l V1 < s s1. Then s1 F r y 1. If T Ž x . l e s B, then g Ž ¨ . - x i for each ¨ g e l Vi , 1 F i F k. Therefore,
Ý g Ž ¨ . - s1 x 1 q Ž r y s1 . y F Ž r y 1 . x 1 q y
¨ ge
s Ž k y 1. x1 q y y Ž k y r . x1 F
k r
y
kyr r
s 1,
and we obtain a contradiction to the definition of g. Now we need to define a probability measure m on B. To this end, let m j be the uniform measure on Bj with m j Ž Bj . s 1rk and let m s Ý kjs1 m j.
APPROXIMATE SET COVERING
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Then m is a probability measure on B. For a given coordinate 1 F i F k, there are k y 1 intervals Bj , for which the marginal distribution m ij is uniform on w0, 1rr x and one interval Bi for which m ii is uniform on w1rr, krŽŽ k y 1. r .x. Hence m i is uniform on the interval w0, 1rr x with m i Žw0, 1rr x. s Ž k y 1.rk and also is uniform on the interval w1rr, krŽŽ k y 1. r .x with m i Žw1rr, krŽŽ k y 1. r .x. s 1rk. Whereas Ž1rr .rŽ krŽŽ k y 1. r . y 1rr . s k y 1, we derive that m i is uniform on the whole interval w0, krŽŽ k y 1. r .x. Therefore the set B and the measure m , required in Lemma 1, have been found, and we apply the lemma to get the desired result. Now we turn the preceding proofs into polynomial time approximation algorithms. Taking a closer look at the first proof, we note that it may proceed under a weaker assumption on a hypergraph H. Namely, it suffices to assume that for every subset V0 : V the induced subhypergraph H w V0 x contains an independent set I of weight at least w Ž V0 .rk Žthis assumption clearly holds true for k-colored hypergraphs.. This observation is utilized by the following recursive algorithm. ALGORITHM A1 Input: An r-uniform hypergraph H s Ž V, E ., a weight function w: V ª Rq, and an algorithm INDŽ H X , w ., returning an independent set I : V X of weight at least w Ž V X .rk in any given induced subhypergraph H X s Ž V X , EX . of H. Output: A cover C of H. 1. C s B; 2. Delete all isolated vertices of H; 3. Find on optimal fractional cover g: V ª Rq; 4. if g Ž ¨ . ) 0 for every ¨ g V begin I s INDŽ H, w .; C s V _ I; end; 5. else begin Find a vertex ¨ 1 g V with g Ž ¨ 1 . G 1rŽ r y 1.; if ¨ 14 is a cover of H, set C s ¨ 1 4 ; else begin Define H X s H w V y ¨ 1 x, wX Ž ¨ . s w Ž ¨ . for every ¨ g V y ¨ 1; C s ¨ 1 4 j A1Ž H X , wX .; end; end; 6. return Ž C .. Proof 1 of Theorem 1 implies immediately the following corollary. COROLLARY 1.
R A1 F max r y 1, ŽŽ k y 1.rk . r 4 .
The obvious drawback of the foregoing algorithm lies in its recursivity, which results in a multiple solution of the LP problem Ž2.. If a proper
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MICHAEL KRIVELEVICH
k-coloring Ž V1 , . . . , Vk . of H is given, we can use another algorithm based on Proof 2 of Theorem 1. A careful examination of Proof 2 reveals the following facts: 1. The set B is a union of k intervals B1 , . . . , Bk , none of which is parallel to any coordinate axis. 2. While moving along each interval Bj and building the corresponding cover T Ž x ., one can see that T Ž x . may change only at the point x s Ž x 1 , . . . , x k . for which there exist an index i, 1 F i F k, and a vertex ¨ g Vi for which g Ž ¨ . s x i . Moreover, given such a pair Ž x, ¨ ., we check if Bj contains a point y s Ž y 1 , . . . , y k . with yi ) x i . If such a point indeed exists, we denote I1 s 1 F i F k: yi ) x i 4 , I2 s w1, k x_ I1. Making an infinitesimally small step along Bj from x toward y and obtaining a new point x q dxg Bj , we notice that T Ž x q dx. s
D ¨ g Vi : g Ž ¨ . ) x i 4 j D ¨ g Vi : g Ž ¨ . G x i 4 .
igI1
igI 2
If x i is the maximal value of the ith coordinate in Bj , we consider the set k
T Ž x. s
D ¨ g Vi : g Ž ¨ . G x i 4 . is1
The preceding two facts show that it suffices to check only a finite number of points from B. ALGORITHM A2 Input: An r-uniform hypergraph H s Ž V, E ., a weight function w: V ª Rq, and a proper k-coloring Ž V1 , . . . , Vk . of H. Output: A cover C of H. 1. Find an optimal fractional cover g: V ª Rq; 2. Define a set B : R k as in Proof 2 of Theorem 1; 3. for each color Vi do for each vertex ¨ g Vi do for each interval Bj of B do if there exists a point x s Ž x 1 , . . . , x k . g Bj with g Ž ¨ . s x i begin if there exists a point y s Ž y 1 , . . . , y k . g Bj with yi ) x i begin I1 s 1 F i F k: yi ) x i 4 ; I2 s 1, . . . , k 4 _ I1; end; else begin I1 s B; I2 s 1, . . . , k 4 ; end; T s Di g I1 ¨ g Vi : g Ž ¨ . ) x i 4 j Di g I 2 ¨ g Vi : g Ž ¨ . G x i 4 ; end;
APPROXIMATE SET COVERING
129
4. C s a subset T : V having the smallest weight among the subsets T found in step 3; 5. return Ž C .. Proof 2 yields the following corollary. COROLLARY 2.
R A2 F max r y 1, ŽŽ k y 1.rk . r 4 .
We will apply algorithm A1 to the set covering problem restricted to r-uniform hypergraphs of bounded maximal degree D ŽSection 4.. Algorithm A2 will be applied in Sections 5 and 6 to a general set covering problem for r-uniform hypergraphs.
4. APPROXIMATE COVERS IN HYPERGRAPHS OF BOUNDED DEGREE Suppose that the family of instances of the set cover problem is restricted to r-uniform hypergraphs of maximal vertex degree at most D. We would like to get an approximation algorithm with approximation ratio better than the trivial ratio r. To apply algorithm A1 of the preceding section, we need an algorithm IND to find an independent set of relatively large weight in r-uniform hypergraphs of maximal degree D. This is provided by the following theorem. THEOREM 2. Let H s Ž V, E . be an r-uniform hypergraph of maximal degree at most D and let w: V ª Rq be a weight function on the ¨ ertices of H. Then H contains an independent set I of weight w Ž I . G ŽŽ r y 1.rr . w Ž V .rD1rŽ ry1.. Moreo¨ er, such an independent set can be found in time polynomial in < V < and < E <. Proof. Define first the weight function we : E ª Rq on the edges of H by setting we Ž e . s ݨ g e w Ž ¨ . for every e g E. Let V0 g V be a random subset of V defined by P w ¨ g V0 x s p. The exact value of p will be chosen later. Denote by X the random variable X s w Ž V0 . ; also denote by Y the random variable
Ys
1 r
Ý egE Ž V 0 .
we Ž e . .
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MICHAEL KRIVELEVICH
By linearity of expectation, EX s
Ý w Ž ¨ . P w ¨ g V0 x s pw Ž V .
¨ gV
and EY s s
1 r pr r
Ý
we Ž e . P e g E Ž V0 . s
egE
Ý Ý wŽ ¨ . s
egE ¨ ge
pr r
1 r
Ý
we Ž e . p r s
egE
Ý w Ž ¨ . dŽ ¨ . F
pr r
Ý
p r Dw Ž V . r
¨ gV
we Ž e .
egE
.
Therefore E w X y Y x G pw Ž V . y
p r Dw Ž V . r
.
Now we choose p s 1rD1rŽ ry1. to maximize the preceding expression. Then Ew X y Y x G
r y 1 wŽ V . r
D1rŽ ry1.
.
Thus there exists a specific set V0 for which the difference X y Y is at least ŽŽ r y 1.rr . w Ž V .rD1rŽ ry1.. Fix such a set V0 and for every edge e g EŽ V0 . delete from V0 a vertex ¨ g e that has the smallest weight w Ž ¨ .. Clearly, the total weight of the deleted vertices does not exceed Ž1rr .Ý e g EŽV . w Ž e . s Y, so the remaining subset I : V0 is independent 0 and has weight at least X y Y G ŽŽ r y 1.rr . w Ž V .rD1rŽ ry1.. The above-described randomized algorithm can be derandomized easily using standard techniques of the conditional expectations method Žsee, e.g., w2, Chap. 15x.. Denote by IND the algorithm described in Theorem 2. Incorporating IND into algorithm A1 of Section 3, we obtain the algorithm B for which the following result holds. COROLLARY 3.
R B F max r y 1, r Ž1 y Ž r y 1.rrD1rŽ ry1. .4 .
5. LOCAL RATIO APPROACH}A GENERAL SCHEME In this section we develop a general scheme of approximation algorithms that combines the local ratio approach of Bar-Yehuda and Even w4x, the idea of Hochbaum w15x based on a coloring argument, and our algorithm A2. Our presentation follows closely that of w4x.
APPROXIMATE SET COVERING
131
LEMMA 2. Let H s Ž V, E . be a hypergraph and let w, w 0 , and w 1 be weight functions on the ¨ ertices of H such that w Ž ¨ . G w 0 Ž ¨ . q w 1Ž ¨ . for e¨ ery ¨ g V. If CU , C0U , and C1U are optimal co¨ ers for the instances Ž H, w ., Ž H, w 0 ., and Ž H, w 1 ., respecti¨ ely, then w Ž CU . G w 0 Ž C0U . q w 1Ž C1U .. Proof. w Ž CU . s G
Ý
wŽ ¨ .
Ý
w 0 Ž ¨ . q w1Ž ¨ .
¨ gC U
¨ gC U
s w 0 Ž CU . q w1 Ž CU . G w 0 Ž C0U . q w 1 Ž C1U . . The last inequality follows from the optimality of the covers C0U and C1U . For a hypergraph H0 define the local ratio lrŽ H0 . of H0 as lrŽ H0 . s < V Ž H0 .
½
wŽ ¨ . y d ,
¨ g V0 ,
wŽ ¨ . ,
otherwise;
4. Run the algorithm A on the instance Ž H, w 0 . to get a cover C for H; 5. return Ž C .. THEOREM 3 ŽThe local ratio theorem.. R LO CALŽ H 0 . Ž H , w . F max lr Ž H0 . , R A Ž H , w 0 . 4 . Proof. Denote R s max lrŽ H0 ., R AŽ H, w 0 .4 . Also, let cU and cU0 be the values of optimal solutions for the instances Ž H, w . and Ž H, w 0 ., respec-
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MICHAEL KRIVELEVICH
tively. Whereas < C l V0 < F < V Ž H0 .<, we have w Ž C . s w 0 Ž C . q d < C l V0 < F w 0 Ž C . q d V Ž H 0 . F R A Ž H , w 0 . cU0 q d lr Ž H0 . cU Ž H0 . F R Ž cU0 q d cU Ž H0 . . F RcU . The last inequality is obtained by defining w 1Ž ¨ . s w Ž ¨ . y w 0 Ž ¨ . and then applying Lemma 2. The algorithm LOCALŽ H0 . can be easily generalized to a family of fixed hypergraphs H0 s H1 , . . . , Ht 4 as follows. ALGORITHM LOCALŽ H0 . Input: A hypergraph H s Ž V, E . with a weight function w: V ª Rq. It is assumed that H0 and A have been fixed in advance. Output: A cover C of H. 1. while for some i there exists a copy of Hi in H with vertex set V0 so that w Ž ¨ . ) 0 for every ¨ g V0 do begin Set d s min ¨ g V 0 w Ž ¨ .; for all ¨ g V0 do w Ž ¨ . s w Ž ¨ . y d ; end; end; 2. C1 s ¨ g V: w s 04 ; 3. V1 s V _C1; 4. run the algorithm A on the instance Ž H w V1 x, w 0 . to obtain a cover C2 for H w V1 x; 5. C s C1 j C2 ; 6. return Ž C .. It should be stressed that to run the preceding algorithm, we should be able to find all copies of the hypergraphs from H0 in H. Clearly, if H0 contains a fixed number of hypergraphs, this can be done in polynomial time, for example, by exhaustive search. Define the local ratio lrŽ H0 . of the family H0 as lrŽ H0 . s max H i g H 0 lrŽ Hi .. COROLLARY 4 ŽThe local ratio corollary.. R LO CALŽ H 0 . Ž H , w . F max lr Ž H0 . , R A Ž H w V1 x , w 0 . 4 . Corollary 4 can be proven easily by induction on the number of iterations of step 2, using the local ratio theorem ŽTheorem 3.. Taking another look at the foregoing algorithm, we note that at the end of step 4, the set V1 does not contain a copy of any hypergraph from H0 .
APPROXIMATE SET COVERING
133
Thus we may expect that the resulting subhypergraph H w V1 x, which serves as input for the algorithm A, is relatively sparse and hence can be efficiently colored by a small number of colors. This observation prompts the use of the approximation algorithm A2 for colored hypergraphs as described in Section 3. The simplest application of the local ratio corollary arises when H0 consists of one hypergraph that is just a single edge. In this case, V1 will clearly span no edge and there will be no need for algorithm A. The algorithm LOCALŽ H0 . then reduces to the following simple procedure: while H contains an edge e g EŽ H . with all vertices of positive weight, choose such an edge e, set d s min ¨ g e w Ž ¨ ., and update w Ž ¨ . s w Ž ¨ . y d for all ¨ g e; if all edges contain vertices of zero weight w Ž ¨ . s 0, return the set of all vertices of weight zero as the output. This algorithm runs in O Ž< E <. steps and has approximation ratio r by the local ratio corollary. 6. LOCAL RATIO APPROACH}IMPLEMENTATION In this section we analyze the performance of the algorithm LOCALŽ H0 . for H0 s H0 4 and for a particular choice of a hypergraph H0 . We assume that the uniformity number r is fixed throughout the section. Define a hypergraph H0 s Ž V, E . as follows. The vertex set V Ž H0 . consists of 2 r y 1 vertices denoted by ¨ 1 , . . . , ¨ 2 ry1. The edge set EŽ H0 . consists of r edges of type Ž ¨ 1 , . . . , ¨ ry1 , ¨ i ., where i ranges from r to 2 r y 1, and one additional edge Ž ¨ r , ¨ rq1 , . . . , ¨ 2 ry1 .. One can see easily that a minimal unweighted cover of H0 has size 2; therefore, H0 has local ratio lrŽ H0 . s Ž2 r y 1.r2 s r y 0.5. Note also that for r s 2 the corresponding hypergraph H0 is simply a triangle. Now we claim that every H0-free hypergraph H contains a relatively large independent set and therefore can be colored with a relatively small number of colors, thus providing a platform for using algorithm A2. LEMMA 3. There exists a constant c s cŽ r . such that if H s Ž V, E . is an r-uniform H0-free hypergraph on n ¨ ertices, then H contains an independent set of size at least cn1r r , which can be found in time polynomial in n. Proof. Let < E < s f. Averaging implies that there exist r y 1 vertices u1 , . . . , u ry1 g V Ž H . such that H contains at least f Ž r yr 1 .rŽ r yn 1 . s rfrŽ r yn 1 . edges passing through u1 , . . . , u ry1. Let e1 , . . . , e s be these edges, s then s G rfrŽ r yn 1 .. Denote U s Dis1 e i _ u1 , . . . , u ry14 . Then < U < s s and U does not span an edge from E Žif such an edge e0 existed, then e0 together with the r edges from e1 , . . . , e s that intersect it would form a copy of H0 .; hence, U is an independent set of size at least rfrŽ r yn 1 ..
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MICHAEL KRIVELEVICH
On the other hand, choosing each vertex ¨ g V to belong to a random subset W of V independently and with probability p s Ž nrrf .1rŽ ry1. Žif nrrf ) 1 the assertion of the lemma is trivial. and calculating expectations as done in the proof of Theorem 2, we can show that H contains an independent set of size at least np y fp r s Ž r y 1.rr Ž n rrrf .1rŽ ry1., which can be found in polynomial time using the method of conditional expectations. Therefore, H contains an independent set of size at least
¡ rf r y 1 n ¢ž r yn 1 / , r ž rf /
~
r
¦ ¥ § G cn
1r Ž ry1 .
1r r
max
for some constant c s cŽ r ., as claimed. An algorithm for finding an independent set is easily converted into a coloring algorithm as follows. At each step, we find an independent set in the current hypergraph, color it by a fresh color, and remove it from the hypergraph. The following lemma, used in most of the papers on approximate graph coloring Žsee, e.g., w16, 26, 12x., provides an upper bound on the number of colors. LEMMA 4. An iterati¨ e application of an algorithm to find an independent set of size cn1r r in a hypergraph H on n ¨ ertices produces a coloring of H with no more than rrŽ cŽ r y 1.. nŽ ry1.r r colors. Proof. Denote by f Ž n. the guaranteed size of the output of the independent set algorithm applied to an H0-free hypergraph H on n vertices. Then f Ž n. G cn1r r. We may assume that f is a continuous, positive and nondecreasing function of its real argument. Let V1 , . . . , Vk be the resulting coloring of the above-described coloring algorithm. We assume that V1 is the fist color used, V2 is the second one, and so on. Enumerate the n vertices of H by the numbers 1, . . . , n in such a way that if i1 g Vj1 and i 2 g Vj 2 and j1 - j2 , then i1 ) i 2 Žthat is, vertices with larger numbers got smaller colors.. It can be seen easily that if i g Vj , then < Vj < G f Ž i .. Therefore k
ks
k
Ý 1s Ý Ý js1
js1 igVj
1 < Vj <
n
F
1
. Ý is1 f Ž i .
This can be estimated from above by the integral n
H0
dt f Ž t.
F
1
dt
n
H c 0
t
1r r
s
r c Ž r y 1.
nŽ ry1.r r .
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APPROXIMATE SET COVERING
Now we can define a specific algoirthm A that can be substituted in LOCALŽ H0 .: given an H0-free r-uniform hypergraph H on n vertices, color H by O Ž nŽ ry1.r r . colors as described previously and then use algorithm A2. Denote the above-described procedure by C and the resulting algorithm by LOCALŽ H0 . q C. Then we have Corollary 5. COROLLARY 5. R LO CALŽ H 0 .qC F max
½
2r y 1
sr 1y
ž
2
, r y 1, r 1 y
Q Ž 1. nŽ ry1.r r
ž
Q Ž 1. nŽ ry1.r r
/5
/
for large ¨ alues of n. The algorithm LOCALŽ H0 . q C can be viewed as an extension of the algorithm COVER2 of w4x, based on Wigderson’s algorithm w26x for coloring a triangle-free graph on n vertices in 2'n colors. Comparing with the randomized algorithm of Peleg, Schechtman, and Wool w23x, we note that for the typical case < EŽ H .< s QŽ< V Ž H .< r ., our algorithm has a better approximation ratio. A possible way to improve the preceding results is to extend the family H0 s H0 4 to a family of hypergraphs with local ratio strictly less than r and then to show that an H0-free hypergraph H contains a larger independent set. Bar-Yehuda and Even w4x took H0 to consist of odd cycles of length at most l, where l is a function of the number n of vertices of the given graph. It is not clear what the analog of an odd cycle in r-uniform hypergraphs is for r G 3 here Žour hypergraph H0 can be viewed as an analog of a triangle. and how to search in polynomial time for subhypergraphs from a family of size growing with n. In any case, this aproach cannot give substantially better results for any choice of a fixed family H0 , as demonstrated in the next section.
7. LOCAL RATIO APPROACH}LIMITATIONS The main result of this section shows that for any choice of a fixed family H0 of excluded r-uniform Ž r G 3. hypergraphs the local ratio approach cannot produce an approximation algorithm with approximation ratio asymptotically better than r. A result of a similar flavor was proven w5x for the case of graphs Ž r s 2.. by Boppana and Halldorsson ´ Let H0 s H1 , . . . , Ht 4 be a fixed family of r-uniform hypergraphs. If we want to plug this family into our general algorithm LOCALŽ H0 . and to
136
MICHAEL KRIVELEVICH
obtain an algorithm with approximation ratio better than r, we should require that lrŽ Hi . - r for every Hi g H0 . So assume this is indeed the case and write lrŽ H0 . s r y e for some fixed 0 - e - r. For an r-uniform hypergraph H s Ž V, E ., where < V < ) r, let
r Ž H . s max X
V :V < V X <)r
eŽ V X . y 1
be the density of H. It is easy to check that if H has no isolated vertices, then r Ž H . G < EŽ H .
Ž 9.
Assume first that H has an edge e0 contained entirely in U1. Then if the subset V _ e0 spans no edges, then Žrecall that U1 consists of vertices of degree 1. H has only one edge e0 , and therefore lrŽ H . s r, contradicting our assumption. Otherwise, we consider the hypergraph H X s H w V _ e0 x. It has n y r vertices, more than one edge, and its minimal cover has one vertex less than a minimal cover of H. Then X
lr Ž H . s
tŽH .
s
nyr
tŽH. y1
-
n
tŽH.
s lr Ž H .
win this section t Ž H . denotes the covering number of H for the unweighted casex. From the definition of r Ž H . we have r Ž H9. F r Ž H .. Thus we obtain a contradiction to the minimality of H.
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APPROXIMATE SET COVERING
If there exists an edge e0 g EŽ H . with < e0 l U1 < s r y 1, then denote
X X ¨ 0 s e 0 _U1 and consider the hypergraph H s H w V _ e0 x. Clearly, < V Ž H .< X X s n y r and t Ž H . G t Ž H . y 1 Žif C is cover of H , then C j ¨ 0 4 is a
cover of H ., and we get a contradiction in a similar way as before. Summarizing the preceding text, we may assume that every edge e g EŽ H . has at most r y 2 vertices in common with U1. Now we are going to show that the set U1 j U2 contains a relatively large independent subset in H. For every 2 F i F r define the edge set Ei ; E by Ei s e g E Ž H . : < e l U2 < s i , < e l U1 < s r y i 4 and also let a i s < Ei <. Then we have r
2 n2 s
Ý d Ž ¨ . G Ý iai .
¨ gU2
Ž 10 .
is2
If there exist two edges e1 , e2 g E2 such that e1 l e2 / B, then let X
¨ 0 g e1 l e2 l U2 . Considering the hypergraph H s H w V _ŽŽ e1 _U2 . j Ž e2 _U2 . j ¨ 0 4.x, we see that < V Ž H X .< s n y Ž2 r y 3. and t Ž H X . G t Ž H . y 1 Ža cover of H can be obtained by adding ¨ 0 to a cover of H X ., and we
again get a contradiction. Thus we may assume that all edges of E2 are pairwise disjoint, implying 2 a2 F n 2 .
Ž 11 .
Multiplying Ž10. by 1r24 and Ž11. by 1r12 and adding, we get a2r4 q a3r8 q Ý ris4 iair24 F nr6; therefore, r
Ý is2
ai 2
i
F
n 6
.
Ž 12 .
Let V0 be a random subset of U2 , obtained by choosing each vertex
¨ g U2 to be in V0 independently and with probability 1r2. Then Ew V0 x s
n 2r2 and the expectation of the number of edges spanned by U1 j V0 is Ý ris2 a ir2 i F nr6 by Ž12., and hence there exists a subset V0 : U2 with < V0 < y eŽU1 j V0 . G n 2r2 y n 2r6 s n 2r3. Fix such a set V0 and delete one vertex from every edge spanned by U1 j V0 . Then the remaining subset united with U1 forms an independent set of size at least n1 q n 2r3. Therefore t Ž H . F n y n1 y n 2r3. Hence the local ratio lrŽ H . satisfies nrŽ n y n1 y n 2r3. F < V Ž H .
n2 3
F 1y
ž
1 rye
/
n.
Ž 13 .
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MICHAEL KRIVELEVICH
Multiplying Ž13. by 3r2 and Ž9. by 1r2 and adding, we have 2 n1 q n 2 F 2 y
ž
3 2Ž r y e .
/
n.
Ž 14 .
From the definition of U1 and U2 we have the estimate on the number of edges: EŽ H . s G s
1 r 1 r 1 r
Ý dŽ ¨ . s
¨ gV
1 r
žÝ
dŽ ¨ . q
¨ gU1
Ý dŽ ¨ . q
¨ gU2
Ý
dŽ ¨ .
¨ gV _ Ž U1jU2 .
/
Ž < U1 < q 2 < U2 < q 3 Ž < V < y < U1 < y < U2 < . . Ž 3n y 2 n1 y n 2 . .
Hence it follows from Ž14. that
rŽ H . G s
EŽ H . VŽ H . 1 r
ž
1q
G
Ž 1rr . Ž 3n y Ž 2 y Ž 3r2 Ž r y e . . . n . n 3
2Ž r y e .
/
s
2Ž r y e . q 3 2rŽ r y e .
and we obtain a contradiction to the assumption about H and thus finish the proof. We deduce immediately the following corollary. COROLLARY 6. If H0 s H1 , . . . , Ht 4 is a family of r-uniform hypergraphs, satisfying lrŽ H0 . F r y e , then r Ž H0 . G Ž2Ž r y e . q 3.rŽ2 r Ž r y e ... Our next step is to prove the following negative Ramsey-type result. LEMMA 6. Let H0 s H1 , . . . , Ht 4 be a fixed family of r-uniform hypergraphs with density r Ž H0 .. Then there exists a constant c s cŽ H0 . such that for e¨ ery sufficiently large integer n there exists an r-uniform hypergraph H0 on n ¨ ertices, ha¨ ing the following properties: H0 does not contain a copy of any hypergraph from H0 ; H0 does not contain an independent set of size u cn1rŽŽ ry1. r Ž H 0 .. Žln n.1rŽ ry1. v. 1. 2.
Proof. This statement can be proven by applying the Lovasz ´ local lemma w7x, as was done in w5x. We choose to present a proof based on using large deviation inequalities, as developed in w19x.
APPROXIMATE SET COVERING
139
For every 1 F i F t let HiX be a subhypergraph of Hi such that r Ž Hi . s Ž< EŽ HiX .< y 1.rŽ< V Ž HiX .< y r .. Setting H0X s H1X , . . . , HtX 4 , note that if H is H0X-free, then it is clearly H0-free; therefore, we may assume that r Ž Hi . s Ž< EŽ Hi .< y 1.rŽ< V Ž Hi .< y r .. For every 1 F i F t set ¨ i s < V Ž Hi .< and f i s < EŽ Hi .<. Set also f min s min f i : 1 F i F t 4 , f max s max f i : 1 F i F t 4 . Clearly we may assume that r Ž H0 . ) 1rŽ r y 1.; otherwise, there is nothing to prove. Consider a random r-uniform hypergraph Hr Ž n, p . }an r-uniform hypergraph with vertex set V of size < V < s n, in which every r-tuple e : V is chosen to be an edge of H independently and with probability p. We set p s c 0 ny1r r Ž H 0 ., where 0 - c 0 - 1 is a sufficiently small constant. Let n 0 s u cn1rŽŽ ry1. r Ž H 0 .. Žln n.1rŽ ry1. v. For every subset V0 ; V of size < V0 < s n 0 , let X V be the random variable, counting the number of edges 0 of H, spanned by V0 . Also, denote by YV 0 the number of subhypergraphs of H, each isomorphic to one of the hypergraphs from H0 and having at least one edge inside V0 , and by Z V 0 the maximal number of pairwise edge disjoint subhypergraphs of H, each isomorphic to one of the hypergraphs from H0 and having at least one edge inside V0 . Clearly, Z V 0 F YV 0 . Denote by AV 0 the event X V 0 ) f max Z V 0 . Claim 1. If AV 0 holds for every V0 ; V of size < V0 < s n 0 , then H contains a subhypergraph H0 on n vetices, satisfying the requirements of the lemma. Proof. Let H be a maximal under inclusion family of pairwise edge disjoint subhypergraphs of H, each isomorphic to one of the hypergraphs from H0 . Deleting all edges of all subhypergraphs from H, we clearly obtain an H0-free hypergraph H0 on n vertices. For a subset V0 ; V of size < V0 < s n 0 , denote by H V 0 the subfamily of H that consists of all hypergraphs from H that have at least one edge inside V0 . From the definition of Z V 0 it follows that
140
MICHAEL KRIVELEVICH
then the probability P w AV 0x is exponentially small, implying in turn that the probability of the existence of a set V0 , for which AV 0 holds, is less than 1. The random variable X V 0 is binomially distributed with parameters Ž nr0 . and p; therefore, well known estimates on the tails of binomial distribution owing to Chernoff Žsee, e.g., w2, Appendix Ax. assert that for every 0 - a - 1, P XV0 - Ž 1 y a .
n0 n p - exp ya 2 0 pr2 . r r
ž ž / /
ž /
Ž 15 .
Now we turn to bounding the upper tail of Z V 0 . The random variable Z V 0 is tightly connected with another random variable YV 0 . Claim 2.
P w Z V 0 G j x F Ž EYV 0 . jrj! for every natural j.
Proof. This is a particular case of the general result of Erdos ˝ and Tetali w8x Žsee also w2, Chap. 8, Lemma 4.1x.. In particular, we deduce from Claim 2 that P Z V 0 G 5EYV 0 F
e
ž /
5 E YV 0
.
5
Ž 16 .
Let us write YV 0 s YV 0 , 1 q ??? qYV 0 , t , where YV 0 , i is the number of copies of Hi that have at least one edge in EŽ V0 .. Representing YV 0 , i as a sum of indicator random variables, we get n0 r
ž /ž
n y n0 n p f i F EYV 0 , i F 0 ¨i y r r
/
ž /ž
nyr
fi ¨i y r ¨i! p .
/
Therefore ci , 1
f y1 f y1 n0 n p Ž nŽ ¨ iyr .rŽ f iy1. . i F EYV 0 , i F c i , 2 0 p Ž nŽ ¨ iyr .rŽ f iy1. . i , r r
ž /
ž /
where c i, 1 and c i, 2 are some positive constants depending only on Hi . The definitions of r Ž H0 . and p imply that c1 c0f ma xy1
n0 n p F EYV 0 F c 2 c 0f mim y1 0 p, r r
ž /
ž /
where c1 s c1Ž H0 . and c 2 s c 2 Ž H0 . are positive constants.
141
APPROXIMATE SET COVERING
Comparing EX V 0 and EYV 0 we observe 1 c 2 c 0f mi n y1
F
EX V 0
F
EYV 0
1 c1 c 0f maxy1
.
Let us choose c 0 so that the expression c 2 c 0f mi ny1 will be equal to, say 1r10 f max . Then 10 f max F
EX V 0 EYV 0
F
c2 c1
ma x qf min 10 f cyf 0 max .
Now, by Ž15. with a s 1r2 and Ž16., P AV 0 s P X V 0 F f max Z V 0 F P X V 0 F F P XV0 F
EX V 0 2
EX V 0 2
q P f max Z V 0 G
EX V 0
q P Z V 0 G 5EYV 0
mi n qf ma x c1 cyf 0 n0 p w 5 ln 5 y 5 x n 0 p F exp y q exp y r 8 r 10c 2 f max
ž ž / /
2
ž
ž /
/
F exp Ž yc3 n 0r p . for some constant c 3 ) 0. Therefore n P 'V0 : AV 0 F n exp Ž yc3 n 0r p . . 0
ž /
Using the inequality
n n0 n 0 F Ž enrn 0 . , we write
ž /
en n r ry1 n 0 exp Ž yc3 n 0 p . F n ? exp Ž yc3 n 0 p . 0
ž /
ž
n0
/
.
Taking c sufficiently large it follows that P wH
1r Ž ry1 ..
.
142
MICHAEL KRIVELEVICH
The final step is to note that the expression 2 r Ž r y e .rŽŽ r y 1.Ž2Ž r y e . q 3.. is always strictly less than 1 for all r G 3 and all 0 - e - r. How should we interpret Corollary 7? Actually, it indicates that the subhypergraph exclusion algorithm LOCALŽ H0 ., described in Section 5, cannot have approximation ratio better than r Ž1 y 1rn d . for any fixed family H0 , where d s d Ž H0 . Žor, more precisely, we cannot show it by our tools of analysis .. Some new ideas and algorithms are needed to make a breakthrough in this important problem.
REFERENCES 1. R. Aharoni, R. Holzman, and M. Krivelevich, On a theorem of Lovasz ´ on covers in r-partite hypergraphs, Combinatorica 16 Ž1996., 149]174. 2. N. Alon and J. H. Spencer, ‘‘The Probabilistic Method,’’ Wiley, New York, 1992. 3. R. Bar-Yehuda and S. Even, A linear time approximation algorithm for the weighted vertex cover problem, J. Algorithms 2 Ž1981., 189]203. 4. R. Bar-Yehuda and S. Even, A local ratio theorem for approximating the weighted vertex cover problem, in ‘‘Analysis and Design of Algorithms for Combinatorial Problems,’’ ŽG. Ausiello and M. Lucertini, Eds.., Annals of Discrete Math., Vol. 25, pp. 27]46, Elsevier, Amsterdam, 1985. 5. R. Boppana and M. M. Halldorsson, Approximating maximum independent sets by ´ excluding subgraphs, Bit 32 Ž1992., 180]196. 6. F. R. K. Chung, Z. Furedi, M. R. Garey, and R. L. Graham, On the fractional covering ¨ number of hypergraphs, SIAM J. Discrete Math. 1 Ž1988., 45]49. 7. P. Erdos ˝ and L. Lovasz, ´ Problems and results on 3-chromatic hypergraphs and some related questions, in ‘‘Infinite and Finite Sets’’ ŽA. Hajnal et al., Eds.., pp. 609]628, North-Holland, Amsterdam, 1975. 8. P. Erdos ˝ and P. Tetali, Representations of integers as the sum of k terms, Random Structures Algorithms 1 Ž1990., 245]261. 9. Z. Furedi, Matchings and covers in hypergraphs, Graphs Combin. 4 Ž1988., 115]206. ¨ 10. M. R. Garey and D. S. Johnson, ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ Freeman, San Francisco, 1979. 11. N. G. Hall and D. S. Hochbaum, A fast approximation for the multicovering problem, Discrete Appl. Math. 15 Ž1986., 35]40. 12. M. M. Halldorsson, A still better performance guarantee for approximate graph coloring, ´ Inform. Process. Lett. 45 Ž1993., 19]23. 13. M. M. Halldorsson and J. Radhakrishnan, Greed is good: Approximating independent ´ sets in sparse and bounded-degree graphs, in ‘‘Proceedings of the 26th Annual ACM Symposium on Theory of Computing,’’ ACM, New York, 1994, pp. 439]448. 14. D. S. Hochbaum, Approximation algorithms for the set covering and vertex cover problems, SIAM J. Comput. 11 Ž1982., 555]556. 15. D. S. Hochbaum, Efficient bounds for the stable set, vertex cover and set packing problems, Discrete Appl. Math. 6 Ž1983., 243]254. 16. D. S. Johnson, Worst case behaviour of graph coloring algorithms, in ‘‘Proceedings of the 5th Southeastern Conference on Combinatorics, Graph Theory and Computing,’’ Congr. Numer. 10, 1974, pp. 512]527. 17. N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 Ž1984., 373]395.
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18. R. Karp, Reducibility among combinatorial problems, in ‘‘Complexity of Computer Computations’’ ŽE. Miller and J. W. Thatcher, Eds.., pp. 85]103, Plenum, New York, 1972. 19. M. Krivelevich, Bounding Ramsey numbers through large deviation inequalities, Random Structures Algorithms 7 Ž1995., 145]155. 20. B. Monien and E. Speckenmeyer, Ramsey numbers and an approximation algorithm for the vertex cover problem, Acta Inform. 22 Ž1985., 115]123. 21. G. L. Nemhauser and L. E. Trotter, Vertex packing: Structural properties and algorithms, Math. Programming 8 Ž1975., 232]248. 22. C. H. Papadimitriou and M. Yannakakis, Optimization, approximation, and complexity classes, J. Comput. System Sci. 43 Ž1991., 425]440. 23. D. Peleg, G. Schechtman, and A. Wool, Approximating bounded 0]1 integer linear programs, in ‘‘Proceedings of the 2nd Israeli Symposium on Theory of Computing and Systems,’’ IEEE Computer Society, New York, 1993, pp. 69]77. 24. P. Raghavan and C. D. Thompson, Randomized rounding: A technique for provably good algorithms and algorithmic proofs, Combinatorica 7 Ž1987., 365]374. 25. J. B. Shearer, A note on the independence number of triangle-free graphs, Discrete Math. 46 Ž1983., 83]87. 26. A. Wigderson, Improving the performance guarantee for approximate graph coloring, J. ACM 30 Ž1983., 729]735.
25, 118]143 Ž1997.
AL970872
Approximate Set Covering in Uniform Hypergraphs Michael KrivelevichU Department of Mathematics, Raymond and Be¨ erly Sackler Faculty of Exact Sciences, Tel-A¨ i¨ Uni¨ ersity, Tel-A¨ i¨ , Israel Received November 29, 1995; revised April 23, 1997
The weighted set covering problem, restricted to the class of r-uniform hypergraphs, is considered. We propose a new approach, based on a recent result of Aharoni, Holzman, and Krivelevich about the ratio of integer and fractional covering numbers in k-colorable r-uniform hypergraphs. This approach, applied to hypergraphs of maximal degree bounded by D, yields an algorithm with approximation ratio r Ž1 y crD1r Ž ry1. .. Next, we combine this approach with an adaptation of the local ratio theorem of Bar-Yehuda and Even for hypergraphs and present a general framework of approximation algorithms, based on subhypergraph exclusion. An application of this scheme is described, providing an algorithm with approximation ratio r Ž1 y crnŽ ry1.r r . for hypergraphs on n vertices. We discuss also the limitations of this approach. Q 1997 Academic Press
1. INTRODUCTION The minimum set co¨ er problem is certainly one of the most central problems of combinatorial optimization. In an instance of this problem, a collection C of subsets of a finite set S and a weight function w: C ª Rq are given, and the task is to find a subset CX : C of minimum weight w Ž CX . s Ý c g C X w Ž c . such that every element of S belongs to at least one member of CX . When all weights equal identically 1, we obtain the unweighted version of the problem. For our purposes it is more convenient to reformulate this problem in terms of hypergraphs. A hypergraph H is an ordered pair H s Ž V, E ., where V is a finite nonempty set Žthe set of ¨ ertices. and E is a collection of distinct nonempty subsets of V Žthe set of edges.. H is called r-uniform * This research forms part of a Ph.D. thesis written by the author under the supervision of Professor Noga Alon. Research supported in part by a Charles Clore Fellowship. 118 0196-6774r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
APPROXIMATE SET COVERING
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if < e < s r for all edges e g EŽ H .. The set cover problem can be represented as follows: for given hypergraph H s Ž V, E . and weight function w: V ª Rq, solve the optimization problem min
Ý wŽ ¨ . g Ž ¨ .
¨ gV
s.t.
Ý gŽ¨. G 1
for every e g E Ž H . ,
¨ ge
g Ž ¨ . g 0, 1 4
for every ¨ g V .
Ž 1.
This formulation leads naturally to the fractional relaxation of the integer problem Ž1.: min
Ý wŽ ¨ . g Ž ¨ .
¨ gV
s.t.
Ý gŽ¨. G 1
for every e g E Ž H . ,
¨ ge
g Ž ¨ . G 0 for every ¨ g V .
Ž 2.
For a hypergraph H s Ž V, E ., any feasible solution g of Ž1. Žalternatively, the set of vertices of weight 1 in a feasible solution g . is called a co¨ er of H with ¨ alue < g < s ݨ g V w Ž ¨ . g Ž ¨ .. An optimal solution of Ž1. is an optimal co¨ er of H, having the co¨ ering number of H wwhich we denote by t Ž H .x as its value. Quite similarly, any feasible solution g of Ž2. is a fractional co¨ er of H; an optimal solution of Ž2. is an optimal fractional co¨ er of H with value t U Ž H ., called the fractional co¨ ering number of H. It is important to note that the problem Ž2. is a linear programming problem that can be solved in time polynomial in < V < and < E < w17x. Clearly, the solution t U Ž H . of the fractional problem Ž2. may serve as a lower bound on the solution t Ž H . of the more difficult integer problem Ž1.. In this paper we consider the case of r-uniform hypergraphs, where r is thought to be a fixed number. The above-defined problem Ž1. turns out to be NP-complete even for the unweighted case and r s 2 Žthat is, for the case of graphs, where this problem is usually called the ¨ ertex co¨ er problem., as shown by Karp w18x. This result motivates studying approximate algorithms for this problem. An approximation algorithm A for problem Ž1. is a polynomial time algorithm which, for a given input instance Ž H, w . of Ž1., produces a feasible solution. The value of the solution obtained by A for an instance H is denoted by AŽ H .. The approximation ratio R A of an approximation algorithm A on a family of instances H is R A s sup AŽ H .rt Ž H .: H g H 4 .
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MICHAEL KRIVELEVICH
The approximation ratio R A can be viewed as a quantitative measure of the quality of an algorithm A. Here we aim to develop efficient approximation algorithms for the class Hr of r-uniform hypergraphs for a given and fixed value of r. The set cover problem remains hard even as an approximation problem. Speaking explicitly, it is MAX-SNP complete as proven in w22x. Moreover, a commonly believed conjecture Žsee, e.g., w15, 23x. states that unless P s NP, there does not exist a polynomial time approximation algorithm A with approximation ratio R A F r y e for any positive constant e . Let us briefly describe known results in the positive direction. We start with the first nontrivial case r s 2. The following very simple algorithm, attributed in w10x to Gavril, can be seen easily to have approximation ratio 2 in the unweighted case. The algorithm proceeds as follows. Given a graph G, find a maximal Žunder inclusion. matching M in G Žfor example, by picking edges greedily. and take all vertices in the edges of M as a cover. This idea can be easily generalized to the case of general r. Several algorithms having approximation ratio 2 for the general Žweighted. case are known; the algorithm of Bar-Yehuda and Even w3x is the simplest of them. Most Žif not all. of the more sophisticated approximation algorithms rely heavily on the theorem of Nemhauser and Trotter w21x, which enables reduction of a given problem to the problem restricted to a class of graphs G satisfying < V Ž G .
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so-called local ratio theorem and proceeds by excluding subgraphs G 0 : G that have large covering number relative to their order. Returning to the set covering problem for r-uniform hypergraphs for general r, we mention the algorithm of Hall and Hochbaum w11x for a general multicovering problem wwhere in Ž1. the first group of restrictions is ݨ g e g Ž ¨ . G bŽ e . for each edge e g E x. This algorithm has approximation ratio r and performs O Žmax< V <, < E <4< V <. iterations. Peleg, Schechtman, and Wool w23x presented several algorithms that have approximation ratio r for the unweighted case. They also described a randomized approximation algorithm that has approximation ratio r Ž1 y Ž crm.1r r ., where m is the number of edges in the hypergraph H and c is some absolute constant. This algorithm first solves the fractional problem Ž2. and then uses the technique of randomized rounding with scaling that was developed by Raghavan and Thompson w24x. Now we describe new results presented in this paper and its structure. In Section 2 we gather some results about fractional covers and matchings that will be used in subsequent sections. A recent theorem of Aharoni, Holzman, and Krivelevich w1x that bounds the ratio between integer and fractional covering numbers in k-colorable r-uniform hypergraphs is presented together with two different proofs in Section 3. Both proofs are algorithmic; the corresponding algorithms are described in the same section. In Section 4 we apply one of these algorithms to the problem of approximating the set covering number for r-uniform hypergraphs of maximal vertex degree bounded by D and obtain an algorithm with approximation ratio r Ž1 y crD1rŽ ry1. .. In Section 5 we combine the second algorithm of Section 3 with the local ratio approach of Bar-Yehuda and Even to give a general scheme of an approximation algorithm parameterized by a family H0 of excluded subhypergraphs. A particular realization for a concrete choice of H0 is presented in Section 6, which yields an algorithm with approximation ratio r Ž1 y crnŽ ry1.r r . for hypergraphs on n vertices. We discuss also the limitations of the local ratio approach in Section 7. We close this section with some notation. Given a hypergraph H s Ž V, E . and a weight function w: V ª Rq, the degree dŽ ¨ . of a vertex ¨ g V is the number of edges of H containing ¨ . For a subset V0 : V, the notation H w V0 x stands for the subhypergraph of H induced by V0 . This subhypergraph has set of edges EŽ V0 .. We denote its cardinality by eŽ V0 . s < EŽ V0 .<. A subset V0 : V is called independent in H if eŽ V0 . s 0. The weight w Ž V0 . of V0 is w Ž V0 . s ݨ g V 0 w Ž ¨ .. For a given hypergraph H0 , a hypergraph H is called H0-free, if H does not contain a Žnot necessarily induced. copy of H0 . Given a family of hypergraphs H0 s H1 , . . . , Ht 4 , H is called H0-free if it is Hi-free for each member Hi of H0 .
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2. FRACTIONAL COVERS AND MATCHINGS IN HYPERGRAPHS In this short section we accumulate some basic facts about fractional covers and matchings to be used in the sequel. Let us formulate the LP problem dual to the previously defined problem Ž2. of determining the fractional covering number of a hypergraph H s Ž V, E .. max
Ý f Ž e. egE
s.t.
Ý f Ž e. F w Ž ¨ .
for every ¨ g V Ž H . ,
e2¨
f Ž e. G 0
for every e g E.
Ž 3.
A feasible solution f of this problem is a fractional matching in H with
¨ alue < f < s Ý e g E f Ž e .. A fractional matching that has the largest possible
value is called optimal and its value is the fractional matching number n U Ž H .. The duality theorem of linear programming applied to the pair of problems Ž2. and Ž3. implies Proposition 1. PROPOSITION 1. For e¨ ery hypergraph H s Ž V, E . and e¨ ery weight function w: V ª Rq, the following statements hold true: 1. t U Ž H . s n U Ž H .. 2. If g: V ª Rq is an optimal fractional co¨ er and f : E ª Rq is an optimal fractional matching, then f Ž e. ) 0
implies
Ý g Ž ¨ . s 1,
¨ ge
gŽ¨. ) 0
implies
Ý f Ž e. s wŽ ¨ . .
Ž 4.
¨ ge
ŽThese are the so-called complementary slackness conditions.. We will make a direct use of the following proposition. PROPOSITION 2. For e¨ ery r-uniform hypergraph H s Ž V, E . and e¨ ery weight function w: V ª Rq the following statements hold true: 1. If g is a fractional co¨ er of H, then for e¨ ery subset V0 : V the function g X : V0 ª Rq, defined by g X Ž ¨ . s g Ž ¨ . for e¨ ery ¨ g V0 Ž that is, g X is the restriction of g to V0 ., is a fractional co¨ er of the hypergraph H w V0 x. 2. If g is an optimal fractional co¨ er of H, satisfying g Ž ¨ . ) 0 for e¨ ery ¨ g V, then t U Ž H . s w Ž V .rr.
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APPROXIMATE SET COVERING
Proof. 1. Follows immediately from the definition of a fractional cover. 2. Let f : E ª Rq be an optimal fractional matching of H with respect to w. Then, by the complementary slackness conditions Ž4., wŽ V . s
Ý wŽ ¨ . s Ý Ý f Ž e. s Ý f Ž e. < e l V <
¨ gV
s
¨ gV e2¨
egE U
Ý f Ž e . r s r Ý f Ž e . s rn Ž H . s rt U Ž H . . egE
egE
w9x for additional The reader is referred to the survey paper by Furedi ¨ results about integer and fractional covers and matchings in hypergraphs.
3. COVERS IN k-COLORED HYPERGRAPHS Given a hypergraph H s Ž V, E ., a partition V s V1 j ??? j Vk is a proper k-coloring of H with colors V1 , . . . , Vk if < e l Vi < - < e < for each e g EŽ H . with < e < G 2 and 1 F i F k; that is, no edge, besides of course singletons, is monochromatic. A hypergraph H is k-colorable if it has a proper k-coloring. Hochbaum w15x suggested the following approach to vertex cover approximation. Given an instance Ž G, w ., first reduce the problem to an instance Ž G 0 , w ., for which t Ž G 0 . G w Ž V Ž G 0 ..r2 by applying the Nemhauser]Trotter algorithm, then color G 0 by k colors V1 , . . . , Vk , and take the complement of a color class Vi , having the maximal weight w Ž Vi ., to be an approximate solution, thus obtaining the approximation ratio 2 y 2rk. Trying to generalize this idea for the case of r-uniform hypergraphs for general r, we need to bypass the lack of the Nemhauser]Trotter type result for r G 3. This can be achieved by using the following recent result of Aharoni, Holzman, and Krivelevich w1x, whose proof can be converted into a polynomial time approximation algorithm. THEOREM 1. Let H s Ž V, E . be an r-uniform k-colorable hypergraph and let w: V ª Rq be a weight function on the ¨ ertices of H. Denote by t Ž H . and by t U Ž H . the co¨ ering and the fractional co¨ ering numbers of H, respecti¨ ely, with respect to w. Then
tŽH. U
t ŽH.
½
F max r y 1,
ky1 k
5
r .
wWe remark that in w1x bounds on the ratio t Ž H .rt U Ž H . are proven for various types of r-uniform hypergraphs.x For the sake of completeness we provide here a proof of the Theorem 1. Actually, we give two different proofs, to be converted later into two approximation algorithms.
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MICHAEL KRIVELEVICH
Proof 1. Suppose that the theorem fails and let H s Ž V, E . with w: V ª Rq be a counterexample with the smallest number of vertices. Then we must have De g E e s V. Fix some proper k-coloring V1 , . . . , Vk of H. Let g: V ª Rq be an optimal fractional cover of H and let f : E ª Rq be an optimal fractional matching in H with respect to w. We distinguish between two cases. Case 1. has
g Ž ¨ . ) 0 for all ¨ g V. Then, by Proposition 2, part 2, one
tUŽ H . s
wŽ V . r
.
Ž 5.
On the other hand, the complement of a heaviest color class is clearly a cover of H with weight not exceeding ŽŽ k y 1.rk . w Ž V .. Therefore,
tŽH. F
ky1 k
wŽ V . .
Ž 6.
Comparing Ž5. and Ž6., we derive t Ž H .rt U Ž H . F ŽŽ k y 1.rk . r}a contradiction with the choice of H. Case 2. There exists a vertex ¨ 0 g V with g Ž ¨ 0 . s 0. By our assumption about H, this vertex belongs to some edge e0 g E. Whereas < e0 < s r and ݨ g e 0 g Ž ¨ . G 1, there exists a vertex ¨ 1 g e0 with g Ž ¨ 1 . G 1rŽ r y 1.. If ¨ 14 is a cover of H, then clearly t Ž H . F w Ž ¨ 1 .. On the other hand, by the complementary slackness conditions Ž4., we have w Ž ¨ 1 . s Ý e 2 ¨ 1 f Ž e .. Therefore, t U Ž H . s n U Ž H . G Ý e 2 ¨ 1 f Ž e . s w Ž ¨ 1 ., so in fact t Ž H . s t U Ž H . s w Ž ¨ 1 ., contradicting the choice of H. So we may consider the hypergraph H X s H w V y ¨ 1 x. Define a new weight function wX : V Ž H X . ª Rq by wX Ž ¨ . s w Ž ¨ . for all ¨ g V Ž H X .. Whereas H X is also r-uniform and k-colorable, it follows from the minimality of H that the theorem statement is valid for H X . Obviously,
t Ž H . F t Ž H . q wŽ ¨1.
Ž 7.
Ža cover of H X can be extended to a cover of H by adding ¨ 1 .. On the other hand, by Proposition 2, part 1, we conclude that
t U Ž H X . F t U Ž H . y g Ž ¨ 1 . w Ž ¨ 1 . F t U Ž H . y w Ž ¨ 1 . r Ž r y 1. . Ž 8.
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APPROXIMATE SET COVERING
It follows from Ž7. and Ž8. that
t Ž H . F t Ž H X . q w Ž ¨ 1 . F max r y 1,
½
½ ½
F max r y 1, F max r y 1,
ky1 k ky1 k
r
ky1
tU H y
5ž Ž . 5 Ž .
k
r t U Ž HX . q wŽ ¨1.
5
wŽ ¨1. ry1
/
q wŽ ¨1.
r tU H ,
again obtaining a contradiction to the choice of H. Proof 2. We present this proof in the following probabilistic setting. LEMMA 1. Let H s Ž V, E . be an r-uniform hypergraph and let w: V ª Rq be a weight function on the ¨ ertices of H. Let g: V ª Rq be an optimal fractional co¨ er of H with respect to w. Suppose V s V1 j ??? j Vk is a partition of V. Suppose further that for some d ) 0 there exists a set B : w0, d x k such that for e¨ ery x s Ž x 1 , . . . , x k . g B the set k
T Ž x. s
D ¨ g Vi : g Ž ¨ . G x i 4 is1
is a co¨ er of H. If there exists a probability measure m defined on B w m Ž B . s 1x such that all marginal distributions m i , 1 F i F k, are uniform on the inter¨ al w0, d x Ž that is, if x g B is randomly chosen from B according to the measure m , then P w a F x i F b x s Ž b y a.rd for e¨ ery 0 F a F b F d ., then
tŽH. U
t ŽH.
F
1
d
.
Proof. Let x g B be randomly chosen from B according to the measure m. Define a random variable Y s w ŽT Ž x .., where T Ž x . is as defined previously. Let us estimate the expectation of Y. By linearity of expectation, Ew Y x s ݨ g V w Ž ¨ . P w Y¨ s 1x, where Y¨ is the indicator random variable for ¨ g V being selected to T. Whereas m has marginal distributions uniform on the interval w0, d x, for every 1 F i F k and for every ¨ g Vi we have P w Y¨ s 1 x s P w ¨ g T x s P g Ž ¨ . G x i s min 1, g Ž ¨ . rd 4 F g Ž ¨ . rd ; hence, Ew Y x s
Ý w Ž ¨ . P w Y¨ s 1x F Ý w Ž ¨ .
¨ gV
¨ gV
gŽ¨.
d
s
tUŽ H . d
.
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MICHAEL KRIVELEVICH
We conclude that there exists at least one point x g B, for which the corresponding cover T Ž x . has weight w ŽT ., satisfying w ŽT . F t U Ž H .rd . Now our strategy is to find, for a k-colored hypergraph H with a given proper k-coloring V1 , . . . , Vk , an appropriate set B and a measure m and then to apply Lemma 1. Consider first the case k - r. In this case we need to prove t Ž H .rt U Ž H . F r y 1. Whereas every k-colorable hypergraph for k F r is also r-colorable, we may assume without loss of generality that H is an r-colorable hypergraph, thus reducing the second case k G r. Assume now that k G r. Define first k q 1 points Q0 , Q1 , . . . , Q k in w0, krŽŽ k y 1. r .x k as Q0 s
ž
ž
1 r
,...,
Q j s 0, . . . ,
1 r
/
, k
Ž k y 1. r
/
,...,0 ,
^ ` _
j s 1, . . . , k.
jth coord.
Now let Bj be the interval in R k joining Q0 and Q j , and let k
Bs
D Bj . js1
Clearly, B : w0, krŽŽ k y 1. r .x k . To check that T Ž x . is a cover for every x g B, consider, for example, the interval B1 s w Q0 Q1 x. It can be seen easily that for each point x s Ž x 1 , . . . , x k . g B1 we have x 1 G 1rr and x 1 G x i for each 2 F i F k, and the last k y 1 coordinates have the same value which we denote by y. The equation Ž k y 1. x 1 q y s krr is satisfied by both endpoints of B1 and hence by every point of B1. Consider an edge e g E and denote < e l V1 < s s1. Then s1 F r y 1. If T Ž x . l e s B, then g Ž ¨ . - x i for each ¨ g e l Vi , 1 F i F k. Therefore,
Ý g Ž ¨ . - s1 x 1 q Ž r y s1 . y F Ž r y 1 . x 1 q y
¨ ge
s Ž k y 1. x1 q y y Ž k y r . x1 F
k r
y
kyr r
s 1,
and we obtain a contradiction to the definition of g. Now we need to define a probability measure m on B. To this end, let m j be the uniform measure on Bj with m j Ž Bj . s 1rk and let m s Ý kjs1 m j.
APPROXIMATE SET COVERING
127
Then m is a probability measure on B. For a given coordinate 1 F i F k, there are k y 1 intervals Bj , for which the marginal distribution m ij is uniform on w0, 1rr x and one interval Bi for which m ii is uniform on w1rr, krŽŽ k y 1. r .x. Hence m i is uniform on the interval w0, 1rr x with m i Žw0, 1rr x. s Ž k y 1.rk and also is uniform on the interval w1rr, krŽŽ k y 1. r .x with m i Žw1rr, krŽŽ k y 1. r .x. s 1rk. Whereas Ž1rr .rŽ krŽŽ k y 1. r . y 1rr . s k y 1, we derive that m i is uniform on the whole interval w0, krŽŽ k y 1. r .x. Therefore the set B and the measure m , required in Lemma 1, have been found, and we apply the lemma to get the desired result. Now we turn the preceding proofs into polynomial time approximation algorithms. Taking a closer look at the first proof, we note that it may proceed under a weaker assumption on a hypergraph H. Namely, it suffices to assume that for every subset V0 : V the induced subhypergraph H w V0 x contains an independent set I of weight at least w Ž V0 .rk Žthis assumption clearly holds true for k-colored hypergraphs.. This observation is utilized by the following recursive algorithm. ALGORITHM A1 Input: An r-uniform hypergraph H s Ž V, E ., a weight function w: V ª Rq, and an algorithm INDŽ H X , w ., returning an independent set I : V X of weight at least w Ž V X .rk in any given induced subhypergraph H X s Ž V X , EX . of H. Output: A cover C of H. 1. C s B; 2. Delete all isolated vertices of H; 3. Find on optimal fractional cover g: V ª Rq; 4. if g Ž ¨ . ) 0 for every ¨ g V begin I s INDŽ H, w .; C s V _ I; end; 5. else begin Find a vertex ¨ 1 g V with g Ž ¨ 1 . G 1rŽ r y 1.; if ¨ 14 is a cover of H, set C s ¨ 1 4 ; else begin Define H X s H w V y ¨ 1 x, wX Ž ¨ . s w Ž ¨ . for every ¨ g V y ¨ 1; C s ¨ 1 4 j A1Ž H X , wX .; end; end; 6. return Ž C .. Proof 1 of Theorem 1 implies immediately the following corollary. COROLLARY 1.
R A1 F max r y 1, ŽŽ k y 1.rk . r 4 .
The obvious drawback of the foregoing algorithm lies in its recursivity, which results in a multiple solution of the LP problem Ž2.. If a proper
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MICHAEL KRIVELEVICH
k-coloring Ž V1 , . . . , Vk . of H is given, we can use another algorithm based on Proof 2 of Theorem 1. A careful examination of Proof 2 reveals the following facts: 1. The set B is a union of k intervals B1 , . . . , Bk , none of which is parallel to any coordinate axis. 2. While moving along each interval Bj and building the corresponding cover T Ž x ., one can see that T Ž x . may change only at the point x s Ž x 1 , . . . , x k . for which there exist an index i, 1 F i F k, and a vertex ¨ g Vi for which g Ž ¨ . s x i . Moreover, given such a pair Ž x, ¨ ., we check if Bj contains a point y s Ž y 1 , . . . , y k . with yi ) x i . If such a point indeed exists, we denote I1 s 1 F i F k: yi ) x i 4 , I2 s w1, k x_ I1. Making an infinitesimally small step along Bj from x toward y and obtaining a new point x q dxg Bj , we notice that T Ž x q dx. s
D ¨ g Vi : g Ž ¨ . ) x i 4 j D ¨ g Vi : g Ž ¨ . G x i 4 .
igI1
igI 2
If x i is the maximal value of the ith coordinate in Bj , we consider the set k
T Ž x. s
D ¨ g Vi : g Ž ¨ . G x i 4 . is1
The preceding two facts show that it suffices to check only a finite number of points from B. ALGORITHM A2 Input: An r-uniform hypergraph H s Ž V, E ., a weight function w: V ª Rq, and a proper k-coloring Ž V1 , . . . , Vk . of H. Output: A cover C of H. 1. Find an optimal fractional cover g: V ª Rq; 2. Define a set B : R k as in Proof 2 of Theorem 1; 3. for each color Vi do for each vertex ¨ g Vi do for each interval Bj of B do if there exists a point x s Ž x 1 , . . . , x k . g Bj with g Ž ¨ . s x i begin if there exists a point y s Ž y 1 , . . . , y k . g Bj with yi ) x i begin I1 s 1 F i F k: yi ) x i 4 ; I2 s 1, . . . , k 4 _ I1; end; else begin I1 s B; I2 s 1, . . . , k 4 ; end; T s Di g I1 ¨ g Vi : g Ž ¨ . ) x i 4 j Di g I 2 ¨ g Vi : g Ž ¨ . G x i 4 ; end;
APPROXIMATE SET COVERING
129
4. C s a subset T : V having the smallest weight among the subsets T found in step 3; 5. return Ž C .. Proof 2 yields the following corollary. COROLLARY 2.
R A2 F max r y 1, ŽŽ k y 1.rk . r 4 .
We will apply algorithm A1 to the set covering problem restricted to r-uniform hypergraphs of bounded maximal degree D ŽSection 4.. Algorithm A2 will be applied in Sections 5 and 6 to a general set covering problem for r-uniform hypergraphs.
4. APPROXIMATE COVERS IN HYPERGRAPHS OF BOUNDED DEGREE Suppose that the family of instances of the set cover problem is restricted to r-uniform hypergraphs of maximal vertex degree at most D. We would like to get an approximation algorithm with approximation ratio better than the trivial ratio r. To apply algorithm A1 of the preceding section, we need an algorithm IND to find an independent set of relatively large weight in r-uniform hypergraphs of maximal degree D. This is provided by the following theorem. THEOREM 2. Let H s Ž V, E . be an r-uniform hypergraph of maximal degree at most D and let w: V ª Rq be a weight function on the ¨ ertices of H. Then H contains an independent set I of weight w Ž I . G ŽŽ r y 1.rr . w Ž V .rD1rŽ ry1.. Moreo¨ er, such an independent set can be found in time polynomial in < V < and < E <. Proof. Define first the weight function we : E ª Rq on the edges of H by setting we Ž e . s ݨ g e w Ž ¨ . for every e g E. Let V0 g V be a random subset of V defined by P w ¨ g V0 x s p. The exact value of p will be chosen later. Denote by X the random variable X s w Ž V0 . ; also denote by Y the random variable
Ys
1 r
Ý egE Ž V 0 .
we Ž e . .
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MICHAEL KRIVELEVICH
By linearity of expectation, EX s
Ý w Ž ¨ . P w ¨ g V0 x s pw Ž V .
¨ gV
and EY s s
1 r pr r
Ý
we Ž e . P e g E Ž V0 . s
egE
Ý Ý wŽ ¨ . s
egE ¨ ge
pr r
1 r
Ý
we Ž e . p r s
egE
Ý w Ž ¨ . dŽ ¨ . F
pr r
Ý
p r Dw Ž V . r
¨ gV
we Ž e .
egE
.
Therefore E w X y Y x G pw Ž V . y
p r Dw Ž V . r
.
Now we choose p s 1rD1rŽ ry1. to maximize the preceding expression. Then Ew X y Y x G
r y 1 wŽ V . r
D1rŽ ry1.
.
Thus there exists a specific set V0 for which the difference X y Y is at least ŽŽ r y 1.rr . w Ž V .rD1rŽ ry1.. Fix such a set V0 and for every edge e g EŽ V0 . delete from V0 a vertex ¨ g e that has the smallest weight w Ž ¨ .. Clearly, the total weight of the deleted vertices does not exceed Ž1rr .Ý e g EŽV . w Ž e . s Y, so the remaining subset I : V0 is independent 0 and has weight at least X y Y G ŽŽ r y 1.rr . w Ž V .rD1rŽ ry1.. The above-described randomized algorithm can be derandomized easily using standard techniques of the conditional expectations method Žsee, e.g., w2, Chap. 15x.. Denote by IND the algorithm described in Theorem 2. Incorporating IND into algorithm A1 of Section 3, we obtain the algorithm B for which the following result holds. COROLLARY 3.
R B F max r y 1, r Ž1 y Ž r y 1.rrD1rŽ ry1. .4 .
5. LOCAL RATIO APPROACH}A GENERAL SCHEME In this section we develop a general scheme of approximation algorithms that combines the local ratio approach of Bar-Yehuda and Even w4x, the idea of Hochbaum w15x based on a coloring argument, and our algorithm A2. Our presentation follows closely that of w4x.
APPROXIMATE SET COVERING
131
LEMMA 2. Let H s Ž V, E . be a hypergraph and let w, w 0 , and w 1 be weight functions on the ¨ ertices of H such that w Ž ¨ . G w 0 Ž ¨ . q w 1Ž ¨ . for e¨ ery ¨ g V. If CU , C0U , and C1U are optimal co¨ ers for the instances Ž H, w ., Ž H, w 0 ., and Ž H, w 1 ., respecti¨ ely, then w Ž CU . G w 0 Ž C0U . q w 1Ž C1U .. Proof. w Ž CU . s G
Ý
wŽ ¨ .
Ý
w 0 Ž ¨ . q w1Ž ¨ .
¨ gC U
¨ gC U
s w 0 Ž CU . q w1 Ž CU . G w 0 Ž C0U . q w 1 Ž C1U . . The last inequality follows from the optimality of the covers C0U and C1U . For a hypergraph H0 define the local ratio lrŽ H0 . of H0 as lrŽ H0 . s < V Ž H0 .
½
wŽ ¨ . y d ,
¨ g V0 ,
wŽ ¨ . ,
otherwise;
4. Run the algorithm A on the instance Ž H, w 0 . to get a cover C for H; 5. return Ž C .. THEOREM 3 ŽThe local ratio theorem.. R LO CALŽ H 0 . Ž H , w . F max lr Ž H0 . , R A Ž H , w 0 . 4 . Proof. Denote R s max lrŽ H0 ., R AŽ H, w 0 .4 . Also, let cU and cU0 be the values of optimal solutions for the instances Ž H, w . and Ž H, w 0 ., respec-
132
MICHAEL KRIVELEVICH
tively. Whereas < C l V0 < F < V Ž H0 .<, we have w Ž C . s w 0 Ž C . q d < C l V0 < F w 0 Ž C . q d V Ž H 0 . F R A Ž H , w 0 . cU0 q d lr Ž H0 . cU Ž H0 . F R Ž cU0 q d cU Ž H0 . . F RcU . The last inequality is obtained by defining w 1Ž ¨ . s w Ž ¨ . y w 0 Ž ¨ . and then applying Lemma 2. The algorithm LOCALŽ H0 . can be easily generalized to a family of fixed hypergraphs H0 s H1 , . . . , Ht 4 as follows. ALGORITHM LOCALŽ H0 . Input: A hypergraph H s Ž V, E . with a weight function w: V ª Rq. It is assumed that H0 and A have been fixed in advance. Output: A cover C of H. 1. while for some i there exists a copy of Hi in H with vertex set V0 so that w Ž ¨ . ) 0 for every ¨ g V0 do begin Set d s min ¨ g V 0 w Ž ¨ .; for all ¨ g V0 do w Ž ¨ . s w Ž ¨ . y d ; end; end; 2. C1 s ¨ g V: w s 04 ; 3. V1 s V _C1; 4. run the algorithm A on the instance Ž H w V1 x, w 0 . to obtain a cover C2 for H w V1 x; 5. C s C1 j C2 ; 6. return Ž C .. It should be stressed that to run the preceding algorithm, we should be able to find all copies of the hypergraphs from H0 in H. Clearly, if H0 contains a fixed number of hypergraphs, this can be done in polynomial time, for example, by exhaustive search. Define the local ratio lrŽ H0 . of the family H0 as lrŽ H0 . s max H i g H 0 lrŽ Hi .. COROLLARY 4 ŽThe local ratio corollary.. R LO CALŽ H 0 . Ž H , w . F max lr Ž H0 . , R A Ž H w V1 x , w 0 . 4 . Corollary 4 can be proven easily by induction on the number of iterations of step 2, using the local ratio theorem ŽTheorem 3.. Taking another look at the foregoing algorithm, we note that at the end of step 4, the set V1 does not contain a copy of any hypergraph from H0 .
APPROXIMATE SET COVERING
133
Thus we may expect that the resulting subhypergraph H w V1 x, which serves as input for the algorithm A, is relatively sparse and hence can be efficiently colored by a small number of colors. This observation prompts the use of the approximation algorithm A2 for colored hypergraphs as described in Section 3. The simplest application of the local ratio corollary arises when H0 consists of one hypergraph that is just a single edge. In this case, V1 will clearly span no edge and there will be no need for algorithm A. The algorithm LOCALŽ H0 . then reduces to the following simple procedure: while H contains an edge e g EŽ H . with all vertices of positive weight, choose such an edge e, set d s min ¨ g e w Ž ¨ ., and update w Ž ¨ . s w Ž ¨ . y d for all ¨ g e; if all edges contain vertices of zero weight w Ž ¨ . s 0, return the set of all vertices of weight zero as the output. This algorithm runs in O Ž< E <. steps and has approximation ratio r by the local ratio corollary. 6. LOCAL RATIO APPROACH}IMPLEMENTATION In this section we analyze the performance of the algorithm LOCALŽ H0 . for H0 s H0 4 and for a particular choice of a hypergraph H0 . We assume that the uniformity number r is fixed throughout the section. Define a hypergraph H0 s Ž V, E . as follows. The vertex set V Ž H0 . consists of 2 r y 1 vertices denoted by ¨ 1 , . . . , ¨ 2 ry1. The edge set EŽ H0 . consists of r edges of type Ž ¨ 1 , . . . , ¨ ry1 , ¨ i ., where i ranges from r to 2 r y 1, and one additional edge Ž ¨ r , ¨ rq1 , . . . , ¨ 2 ry1 .. One can see easily that a minimal unweighted cover of H0 has size 2; therefore, H0 has local ratio lrŽ H0 . s Ž2 r y 1.r2 s r y 0.5. Note also that for r s 2 the corresponding hypergraph H0 is simply a triangle. Now we claim that every H0-free hypergraph H contains a relatively large independent set and therefore can be colored with a relatively small number of colors, thus providing a platform for using algorithm A2. LEMMA 3. There exists a constant c s cŽ r . such that if H s Ž V, E . is an r-uniform H0-free hypergraph on n ¨ ertices, then H contains an independent set of size at least cn1r r , which can be found in time polynomial in n. Proof. Let < E < s f. Averaging implies that there exist r y 1 vertices u1 , . . . , u ry1 g V Ž H . such that H contains at least f Ž r yr 1 .rŽ r yn 1 . s rfrŽ r yn 1 . edges passing through u1 , . . . , u ry1. Let e1 , . . . , e s be these edges, s then s G rfrŽ r yn 1 .. Denote U s Dis1 e i _ u1 , . . . , u ry14 . Then < U < s s and U does not span an edge from E Žif such an edge e0 existed, then e0 together with the r edges from e1 , . . . , e s that intersect it would form a copy of H0 .; hence, U is an independent set of size at least rfrŽ r yn 1 ..
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MICHAEL KRIVELEVICH
On the other hand, choosing each vertex ¨ g V to belong to a random subset W of V independently and with probability p s Ž nrrf .1rŽ ry1. Žif nrrf ) 1 the assertion of the lemma is trivial. and calculating expectations as done in the proof of Theorem 2, we can show that H contains an independent set of size at least np y fp r s Ž r y 1.rr Ž n rrrf .1rŽ ry1., which can be found in polynomial time using the method of conditional expectations. Therefore, H contains an independent set of size at least
¡ rf r y 1 n ¢ž r yn 1 / , r ž rf /
~
r
¦ ¥ § G cn
1r Ž ry1 .
1r r
max
for some constant c s cŽ r ., as claimed. An algorithm for finding an independent set is easily converted into a coloring algorithm as follows. At each step, we find an independent set in the current hypergraph, color it by a fresh color, and remove it from the hypergraph. The following lemma, used in most of the papers on approximate graph coloring Žsee, e.g., w16, 26, 12x., provides an upper bound on the number of colors. LEMMA 4. An iterati¨ e application of an algorithm to find an independent set of size cn1r r in a hypergraph H on n ¨ ertices produces a coloring of H with no more than rrŽ cŽ r y 1.. nŽ ry1.r r colors. Proof. Denote by f Ž n. the guaranteed size of the output of the independent set algorithm applied to an H0-free hypergraph H on n vertices. Then f Ž n. G cn1r r. We may assume that f is a continuous, positive and nondecreasing function of its real argument. Let V1 , . . . , Vk be the resulting coloring of the above-described coloring algorithm. We assume that V1 is the fist color used, V2 is the second one, and so on. Enumerate the n vertices of H by the numbers 1, . . . , n in such a way that if i1 g Vj1 and i 2 g Vj 2 and j1 - j2 , then i1 ) i 2 Žthat is, vertices with larger numbers got smaller colors.. It can be seen easily that if i g Vj , then < Vj < G f Ž i .. Therefore k
ks
k
Ý 1s Ý Ý js1
js1 igVj
1 < Vj <
n
F
1
. Ý is1 f Ž i .
This can be estimated from above by the integral n
H0
dt f Ž t.
F
1
dt
n
H c 0
t
1r r
s
r c Ž r y 1.
nŽ ry1.r r .
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APPROXIMATE SET COVERING
Now we can define a specific algoirthm A that can be substituted in LOCALŽ H0 .: given an H0-free r-uniform hypergraph H on n vertices, color H by O Ž nŽ ry1.r r . colors as described previously and then use algorithm A2. Denote the above-described procedure by C and the resulting algorithm by LOCALŽ H0 . q C. Then we have Corollary 5. COROLLARY 5. R LO CALŽ H 0 .qC F max
½
2r y 1
sr 1y
ž
2
, r y 1, r 1 y
Q Ž 1. nŽ ry1.r r
ž
Q Ž 1. nŽ ry1.r r
/5
/
for large ¨ alues of n. The algorithm LOCALŽ H0 . q C can be viewed as an extension of the algorithm COVER2 of w4x, based on Wigderson’s algorithm w26x for coloring a triangle-free graph on n vertices in 2'n colors. Comparing with the randomized algorithm of Peleg, Schechtman, and Wool w23x, we note that for the typical case < EŽ H .< s QŽ< V Ž H .< r ., our algorithm has a better approximation ratio. A possible way to improve the preceding results is to extend the family H0 s H0 4 to a family of hypergraphs with local ratio strictly less than r and then to show that an H0-free hypergraph H contains a larger independent set. Bar-Yehuda and Even w4x took H0 to consist of odd cycles of length at most l, where l is a function of the number n of vertices of the given graph. It is not clear what the analog of an odd cycle in r-uniform hypergraphs is for r G 3 here Žour hypergraph H0 can be viewed as an analog of a triangle. and how to search in polynomial time for subhypergraphs from a family of size growing with n. In any case, this aproach cannot give substantially better results for any choice of a fixed family H0 , as demonstrated in the next section.
7. LOCAL RATIO APPROACH}LIMITATIONS The main result of this section shows that for any choice of a fixed family H0 of excluded r-uniform Ž r G 3. hypergraphs the local ratio approach cannot produce an approximation algorithm with approximation ratio asymptotically better than r. A result of a similar flavor was proven w5x for the case of graphs Ž r s 2.. by Boppana and Halldorsson ´ Let H0 s H1 , . . . , Ht 4 be a fixed family of r-uniform hypergraphs. If we want to plug this family into our general algorithm LOCALŽ H0 . and to
136
MICHAEL KRIVELEVICH
obtain an algorithm with approximation ratio better than r, we should require that lrŽ Hi . - r for every Hi g H0 . So assume this is indeed the case and write lrŽ H0 . s r y e for some fixed 0 - e - r. For an r-uniform hypergraph H s Ž V, E ., where < V < ) r, let
r Ž H . s max X
V :V < V X <)r
eŽ V X . y 1
be the density of H. It is easy to check that if H has no isolated vertices, then r Ž H . G < EŽ H .
Ž 9.
Assume first that H has an edge e0 contained entirely in U1. Then if the subset V _ e0 spans no edges, then Žrecall that U1 consists of vertices of degree 1. H has only one edge e0 , and therefore lrŽ H . s r, contradicting our assumption. Otherwise, we consider the hypergraph H X s H w V _ e0 x. It has n y r vertices, more than one edge, and its minimal cover has one vertex less than a minimal cover of H. Then X
lr Ž H . s
tŽH .
s
nyr
tŽH. y1
-
n
tŽH.
s lr Ž H .
win this section t Ž H . denotes the covering number of H for the unweighted casex. From the definition of r Ž H . we have r Ž H9. F r Ž H .. Thus we obtain a contradiction to the minimality of H.
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APPROXIMATE SET COVERING
If there exists an edge e0 g EŽ H . with < e0 l U1 < s r y 1, then denote
X X ¨ 0 s e 0 _U1 and consider the hypergraph H s H w V _ e0 x. Clearly, < V Ž H .< X X s n y r and t Ž H . G t Ž H . y 1 Žif C is cover of H , then C j ¨ 0 4 is a
cover of H ., and we get a contradiction in a similar way as before. Summarizing the preceding text, we may assume that every edge e g EŽ H . has at most r y 2 vertices in common with U1. Now we are going to show that the set U1 j U2 contains a relatively large independent subset in H. For every 2 F i F r define the edge set Ei ; E by Ei s e g E Ž H . : < e l U2 < s i , < e l U1 < s r y i 4 and also let a i s < Ei <. Then we have r
2 n2 s
Ý d Ž ¨ . G Ý iai .
¨ gU2
Ž 10 .
is2
If there exist two edges e1 , e2 g E2 such that e1 l e2 / B, then let X
¨ 0 g e1 l e2 l U2 . Considering the hypergraph H s H w V _ŽŽ e1 _U2 . j Ž e2 _U2 . j ¨ 0 4.x, we see that < V Ž H X .< s n y Ž2 r y 3. and t Ž H X . G t Ž H . y 1 Ža cover of H can be obtained by adding ¨ 0 to a cover of H X ., and we
again get a contradiction. Thus we may assume that all edges of E2 are pairwise disjoint, implying 2 a2 F n 2 .
Ž 11 .
Multiplying Ž10. by 1r24 and Ž11. by 1r12 and adding, we get a2r4 q a3r8 q Ý ris4 iair24 F nr6; therefore, r
Ý is2
ai 2
i
F
n 6
.
Ž 12 .
Let V0 be a random subset of U2 , obtained by choosing each vertex
¨ g U2 to be in V0 independently and with probability 1r2. Then Ew V0 x s
n 2r2 and the expectation of the number of edges spanned by U1 j V0 is Ý ris2 a ir2 i F nr6 by Ž12., and hence there exists a subset V0 : U2 with < V0 < y eŽU1 j V0 . G n 2r2 y n 2r6 s n 2r3. Fix such a set V0 and delete one vertex from every edge spanned by U1 j V0 . Then the remaining subset united with U1 forms an independent set of size at least n1 q n 2r3. Therefore t Ž H . F n y n1 y n 2r3. Hence the local ratio lrŽ H . satisfies nrŽ n y n1 y n 2r3. F < V Ž H .
n2 3
F 1y
ž
1 rye
/
n.
Ž 13 .
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MICHAEL KRIVELEVICH
Multiplying Ž13. by 3r2 and Ž9. by 1r2 and adding, we have 2 n1 q n 2 F 2 y
ž
3 2Ž r y e .
/
n.
Ž 14 .
From the definition of U1 and U2 we have the estimate on the number of edges: EŽ H . s G s
1 r 1 r 1 r
Ý dŽ ¨ . s
¨ gV
1 r
žÝ
dŽ ¨ . q
¨ gU1
Ý dŽ ¨ . q
¨ gU2
Ý
dŽ ¨ .
¨ gV _ Ž U1jU2 .
/
Ž < U1 < q 2 < U2 < q 3 Ž < V < y < U1 < y < U2 < . . Ž 3n y 2 n1 y n 2 . .
Hence it follows from Ž14. that
rŽ H . G s
EŽ H . VŽ H . 1 r
ž
1q
G
Ž 1rr . Ž 3n y Ž 2 y Ž 3r2 Ž r y e . . . n . n 3
2Ž r y e .
/
s
2Ž r y e . q 3 2rŽ r y e .
and we obtain a contradiction to the assumption about H and thus finish the proof. We deduce immediately the following corollary. COROLLARY 6. If H0 s H1 , . . . , Ht 4 is a family of r-uniform hypergraphs, satisfying lrŽ H0 . F r y e , then r Ž H0 . G Ž2Ž r y e . q 3.rŽ2 r Ž r y e ... Our next step is to prove the following negative Ramsey-type result. LEMMA 6. Let H0 s H1 , . . . , Ht 4 be a fixed family of r-uniform hypergraphs with density r Ž H0 .. Then there exists a constant c s cŽ H0 . such that for e¨ ery sufficiently large integer n there exists an r-uniform hypergraph H0 on n ¨ ertices, ha¨ ing the following properties: H0 does not contain a copy of any hypergraph from H0 ; H0 does not contain an independent set of size u cn1rŽŽ ry1. r Ž H 0 .. Žln n.1rŽ ry1. v. 1. 2.
Proof. This statement can be proven by applying the Lovasz ´ local lemma w7x, as was done in w5x. We choose to present a proof based on using large deviation inequalities, as developed in w19x.
APPROXIMATE SET COVERING
139
For every 1 F i F t let HiX be a subhypergraph of Hi such that r Ž Hi . s Ž< EŽ HiX .< y 1.rŽ< V Ž HiX .< y r .. Setting H0X s H1X , . . . , HtX 4 , note that if H is H0X-free, then it is clearly H0-free; therefore, we may assume that r Ž Hi . s Ž< EŽ Hi .< y 1.rŽ< V Ž Hi .< y r .. For every 1 F i F t set ¨ i s < V Ž Hi .< and f i s < EŽ Hi .<. Set also f min s min f i : 1 F i F t 4 , f max s max f i : 1 F i F t 4 . Clearly we may assume that r Ž H0 . ) 1rŽ r y 1.; otherwise, there is nothing to prove. Consider a random r-uniform hypergraph Hr Ž n, p . }an r-uniform hypergraph with vertex set V of size < V < s n, in which every r-tuple e : V is chosen to be an edge of H independently and with probability p. We set p s c 0 ny1r r Ž H 0 ., where 0 - c 0 - 1 is a sufficiently small constant. Let n 0 s u cn1rŽŽ ry1. r Ž H 0 .. Žln n.1rŽ ry1. v. For every subset V0 ; V of size < V0 < s n 0 , let X V be the random variable, counting the number of edges 0 of H, spanned by V0 . Also, denote by YV 0 the number of subhypergraphs of H, each isomorphic to one of the hypergraphs from H0 and having at least one edge inside V0 , and by Z V 0 the maximal number of pairwise edge disjoint subhypergraphs of H, each isomorphic to one of the hypergraphs from H0 and having at least one edge inside V0 . Clearly, Z V 0 F YV 0 . Denote by AV 0 the event X V 0 ) f max Z V 0 . Claim 1. If AV 0 holds for every V0 ; V of size < V0 < s n 0 , then H contains a subhypergraph H0 on n vetices, satisfying the requirements of the lemma. Proof. Let H be a maximal under inclusion family of pairwise edge disjoint subhypergraphs of H, each isomorphic to one of the hypergraphs from H0 . Deleting all edges of all subhypergraphs from H, we clearly obtain an H0-free hypergraph H0 on n vertices. For a subset V0 ; V of size < V0 < s n 0 , denote by H V 0 the subfamily of H that consists of all hypergraphs from H that have at least one edge inside V0 . From the definition of Z V 0 it follows that
140
MICHAEL KRIVELEVICH
then the probability P w AV 0x is exponentially small, implying in turn that the probability of the existence of a set V0 , for which AV 0 holds, is less than 1. The random variable X V 0 is binomially distributed with parameters Ž nr0 . and p; therefore, well known estimates on the tails of binomial distribution owing to Chernoff Žsee, e.g., w2, Appendix Ax. assert that for every 0 - a - 1, P XV0 - Ž 1 y a .
n0 n p - exp ya 2 0 pr2 . r r
ž ž / /
ž /
Ž 15 .
Now we turn to bounding the upper tail of Z V 0 . The random variable Z V 0 is tightly connected with another random variable YV 0 . Claim 2.
P w Z V 0 G j x F Ž EYV 0 . jrj! for every natural j.
Proof. This is a particular case of the general result of Erdos ˝ and Tetali w8x Žsee also w2, Chap. 8, Lemma 4.1x.. In particular, we deduce from Claim 2 that P Z V 0 G 5EYV 0 F
e
ž /
5 E YV 0
.
5
Ž 16 .
Let us write YV 0 s YV 0 , 1 q ??? qYV 0 , t , where YV 0 , i is the number of copies of Hi that have at least one edge in EŽ V0 .. Representing YV 0 , i as a sum of indicator random variables, we get n0 r
ž /ž
n y n0 n p f i F EYV 0 , i F 0 ¨i y r r
/
ž /ž
nyr
fi ¨i y r ¨i! p .
/
Therefore ci , 1
f y1 f y1 n0 n p Ž nŽ ¨ iyr .rŽ f iy1. . i F EYV 0 , i F c i , 2 0 p Ž nŽ ¨ iyr .rŽ f iy1. . i , r r
ž /
ž /
where c i, 1 and c i, 2 are some positive constants depending only on Hi . The definitions of r Ž H0 . and p imply that c1 c0f ma xy1
n0 n p F EYV 0 F c 2 c 0f mim y1 0 p, r r
ž /
ž /
where c1 s c1Ž H0 . and c 2 s c 2 Ž H0 . are positive constants.
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APPROXIMATE SET COVERING
Comparing EX V 0 and EYV 0 we observe 1 c 2 c 0f mi n y1
F
EX V 0
F
EYV 0
1 c1 c 0f maxy1
.
Let us choose c 0 so that the expression c 2 c 0f mi ny1 will be equal to, say 1r10 f max . Then 10 f max F
EX V 0 EYV 0
F
c2 c1
ma x qf min 10 f cyf 0 max .
Now, by Ž15. with a s 1r2 and Ž16., P AV 0 s P X V 0 F f max Z V 0 F P X V 0 F F P XV0 F
EX V 0 2
EX V 0 2
q P f max Z V 0 G
EX V 0
q P Z V 0 G 5EYV 0
mi n qf ma x c1 cyf 0 n0 p w 5 ln 5 y 5 x n 0 p F exp y q exp y r 8 r 10c 2 f max
ž ž / /
2
ž
ž /
/
F exp Ž yc3 n 0r p . for some constant c 3 ) 0. Therefore n P 'V0 : AV 0 F n exp Ž yc3 n 0r p . . 0
ž /
Using the inequality
n n0 n 0 F Ž enrn 0 . , we write
ž /
en n r ry1 n 0 exp Ž yc3 n 0 p . F n ? exp Ž yc3 n 0 p . 0
ž /
ž
n0
/
.
Taking c sufficiently large it follows that P wH
1r Ž ry1 ..
.
142
MICHAEL KRIVELEVICH
The final step is to note that the expression 2 r Ž r y e .rŽŽ r y 1.Ž2Ž r y e . q 3.. is always strictly less than 1 for all r G 3 and all 0 - e - r. How should we interpret Corollary 7? Actually, it indicates that the subhypergraph exclusion algorithm LOCALŽ H0 ., described in Section 5, cannot have approximation ratio better than r Ž1 y 1rn d . for any fixed family H0 , where d s d Ž H0 . Žor, more precisely, we cannot show it by our tools of analysis .. Some new ideas and algorithms are needed to make a breakthrough in this important problem.
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