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A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem
Annals of Discrete Mathematics 25 (1985) 27-46 0 Elsevier Science Publishers B.V. (North-Holland)
27
A LOCAL-RATIO THEOREM FOR APPROXIMATING THE WEIGHTED VERTEX COVER PROBLEM
R. BAR-YEHUDA and S. EVEN Computer Science Deportment Technion - Istroel Institute of
Technology
Hoifo, Istroel32000
A local-ratio theorem for approximating the weighted vertex cover problem is presented. It consists of reducing the weights of vertices in certain subgraphs and has the effect of locdaiproximetion. Putting together the Nemhauser-Trotter local optimization algorithm and the localratio theorem yields several new approximation techniques which improve known results from time complexity, simplicity and performanceratio point of view. The main approximation algorithm guarantees a ratio of 2-- 1 where K is the smalloglog?l K lest integer s.t. (2K-1)K2 n (hence: ratio 5 2 1
210gn
t
This is M improvement over the currently known ratios, especially for a “practical” number of vertices (e.g. for graphs which have less than 2400,60000, 1Ol2 vertices the ratio is bounded by 1.75, 1.8, 1.9 respectively).
1 . Introduction A Vertex Cover of a graph is a subset of vertices such that each edge has at least one endpoint in the subset. The Weighted Vertex Cover Problem (WVC) is defined as follows: Given a simple graph G(V, E ) and a weight function o :V + R’, find a cover of minimum total weight. WVC is .known to be NP-Hard, even if all weigts are 1 [ 161 and the graph is planar [81. Therefore, it is natural to look for efficient approximation algorithms.
Let A be an approximation algorithm. For graph G with weight functions w , let C, , C* be the cover A produces and an optimum cover, f All log bases, in this paper, me 2.
28
R Bar- Yehuda and S, Even
respectively . Define R A (G, W ) =
w
fcA)
w
(C*)
and let the performance ratio r A f n ) be
rA(n) = Sup [ R A ~ G w , ) I G = f V, E l where n = I VI] Many approximation algorithms with performance ratio I 2 have been suggested; see, for example Table 1 . No polynomial-time approximation algorithm A is known for which rAfn) 5 2 - c , where E > 0 and fixed. Several approximation algorithms are known for which R A ~ G ,w ) 5 2- EfG), where E depends on G ; e.g. E (C) =- 2
a/ C)
where AfG) is the maximum degree of the vertices of G .
In Section 2 we review the Nemhauser and Trotter [I81 local optimization algorithm f N T ) as well as the observation of Hochbaum [ 1 I I to use it for approximating WVC. In Section 3, a new theorem, the ‘Local-Ratio Theorem’ is presented and proved. It consists of reducing the weights of vertices in certain subgraphs and has the effect of local approximation. As an example of its power we present a trivial proof for the correctness of a linear time approximation algorithm CO VER 1 with ‘COVER l f n ) 1 2 . In Section 4, we present two approximation algorithms in which the NT algorithm and the local-ratio theorem are shown t o be useful.
The first algorithm COVER2 satisfies r ~ 0 v E R 2 f n )<2--for 1
6
ge-
neral graphs, while for planar graphs r c o V E R 2 f n ) < I .5 and its time complexity is O(n2 logn). Hochbaum [121 obtained the same performance ratio, but we manage to avoid the time complexity of 4-coloring. For our last algorithm, co VER3, we prove that rco VER3fn)5 1 - -,1 k where k is the least integer s.t. (2k-1) k 2: n . A similar result, but only for unweighted graphs, has been obtained independently
A local-ratio theorem
29
by Monien and Speckenrneyer [ 17 I.
In Section 5 , the Local-Ratio Theorem is extended t o the Weighted Set-Cover Problem. I
TABLE 1 -SUMMARY OFAPPROXIMATIONRESULTS. (for NT, see Table 2) Performance ratio I
References
Complexity for weighted
Complexity for un weighted
2
2 2 2--
I
E 2
A L
this paper
2--
this paper
2 - loglog v 210g v
fl
“NT”
“NT”
I
“NT”
I
“NT”
“NT”
EV
(For planar graphs)
I
I
I
thispaper
1.6
“NT”
1.5
“NT” -k
1.5
1.5
“NT” -k
“4- COLORING”
“4 -COLORING”
“NT”
“NT”
not applicable 1 +F
* Although this result seems much better then the one shown 2 lines above for unweighted graphs, the algorithm is not useful for practical computations. For example for the akorithm to achieve a ratio I2 one may need ad many as 2 see [2].
160
vertices [ 6 ].For a similar algorithm
R Bar- Yehuda and S. Even
30
G
=(V, E )
Weighted
Unweighted
E2 BV5 B
General
Planar
or EVlog V V2log v
E f l 1.5
2. The Nemhauser and Trotter local optimization algorithm In this short section we review the local optimization algorithm of Nemhauser and Trotter 1181 which is very useful for approximations of wvc t121. Let G(V, El be a simple graph. We denote by G(U) the subgraph of G induced by U c V and let U‘ =( u’lu E U).Define the weights of vertices in U’by d u ’ ) = ~ ( u ) .
The following theorem states results of Nemhauser and Trotter.
The NT-Theorem: The sets C,, Vo which Algorithm NT produces, satisfie the following properties:
A locabratio theorem
(i)
If a set D
31
c V o covers G ( V o ) then C = D U Co covers G.
(ii) There exists an optimum cover C*(G) such that C*(C) 2 Co. [(i) and (ii) are called the local optimally conditions.] (iii) w(C*fCf V o ) ) ) 2 1/2 d V 0 ) . For a proof see [181. An alternate proof is given in the APPENDIX. The significance of the NT-Theorem, as pointed out by Hochbaum 1121, is that the problem of approximating WVC can be limited to graphs G( V o ) which satisfy (iii). A simple illustration of this approach was used by Hochbaum to obtain an algorithm with performace ratio 1 2 : Call NT(C( V h w ) to get Co, V o and return C = Co U Vo . Let us consider now the time-complexity of finding C*(B). which determines the time-complexity on NT.
For the unweighted case the problem can be converted into the maximum matching problem on B (see for example [31 which is of timecomplexity O ( E P ) (see [141). For the weighted case the problem can be converted into a maximum flow problem (see, for example [12]) which is of time-com[I01 or O(EVlogV) 1191. For a summary of the plexity OfE2'3 results, see Table 2.
3. The local-ratio theorem In a previous paper [21 we presented a local approximation technique for the vertex cover problem of unweighted graphs. In this section we present a local approximation technique for the vertex cover problem of weighted graphs. First, we present the following lemma:
Lemma: Let C( V, El be a graph and w , w and o1 be weight functions on V, s.t. for every Y E V: W ( Y ) 2 w , ( v ) + w l ( v ) . Let c*,C: and C: be optimum covers of C with respect t o a,w and a ? . It follows that: w f C * ) 2 w (C*l +02 (Ct).
R Bar- Yehuda and S. Even
32
= O,fC*)+W2fC*)
2 w 1 f C * ) + w 2 f C * ) [by the optimality of Cr and C,*]
Q.E.D.
c
Let be an unweighted graph of fi vertices, whose optimum cover contains C* vertices. Define F=ii/Z*. Let A be an approximation algorithm for WVC and let L O C A L c be the following algorithm:
Algorithm LOCAL Input:
Graph Gf V , E), with weight function w .
Phase I :
Choose a subgraph G f V, E ) of G which is isomorphic to
N
N
hr
c.
Choose 0 565 Min [wfx) Define w o fx) =
I
X E
V] h
wfx)-6
if x E V
wfx)
else.
Phase 2:
Call A f G , w o ) to get Co.
output:
c + c,.
The Local-Ratio Theorem: R L O C A L ~ ~ G w ,) < M a x [F, RA(G, w o ) ] Proof Let c* and c r be the weights of the optimum covers of G with respect to w and w o , and let r = M a x [ i , R A ~ G w, , ) ] , then fC) I
W,
fC)+S.
n
CI
[by I C n V l l n ]
A locabratio theorem
I RA(G, w o ) . co * +i=.6 . C*
33
[by definitions]
[by r's definition]
Ir . ( c o * + 6 P )
< r . c*
[by the lemma]
Q.E.D. Let us consider now a corollary of the Local-Ratio Theorem. Let I' be a finite family of graphs, and rr = MUXIT I C E r}. We denote by G(U), U G V , the subgraph of C(V, E ) induced by U.
Algorithm LOCAL r Input:
G(V, El, w .
Phase 0:
For every x
Phase 1:
For every G( V, some E I', do
- - z), E
V do w (XI + w (XI end subgraph of G which is isomorphic to
6 +Min[wo (X)IXET).
For every x E
7 do w o (x) +-w o(x) -6
end
end
Phase 2:
c1 +[x I w o ( x ) = 01. v1 v-c1 . +
Call A ( G ( V , ), wo) to get Cz.
ou tpu t:
c+c1 u c,.
The Local-Ratio Corollary: R L O C A L ~(C,0 )S Max
[ r r , RA(G(V,), w O ) } .
Proof: By induction on i, the number of iterations of Phase 1, during which 6 > 0.
R Bar- Yehuda and S. Even
34
For i=O the claim is trivial.
+
Suppose the claim holds for i, and for some G, w Phase 1 runs i 1 iterations during which 6 > 0. Let E r be the graph used in the first iteration during which 6 > 0. Observe that we can view the running of LOCAL, as an application of L O C A L c where A (in Phase 2 of LOC A L c ) is replaced by the remaining part of LOCALr. The inductive step is now an immediate consequence of the Local-Ratio Theorem.
Q.E.D. In the applications of LOCALr we shall refer to rr as (a bound on) the local-ratio of Phase 1. Let us demonstrate a simple application of the Local-Ratio Corollary.
Algorithm CO VER 1
Input:
G(V, E), a.
Phase 1:
For every e E E do Let 6 = Min { w ( x E~ e). ~
For every x
Ee
do w (x) = w(x)-6 end
end Output:
c + [ x I d x ) = 01.
This approximation algorithm is essentially the one we described in ill'.
Proposition: For algorithm COVER 1.
(1) The time complexity is O(E)
(2) Its performance ratio 1 2 Proof: (1)
'
The number of operations spent on each edge is bounded by a
I n [I ] w e used a global rather then a local point of vim.
35
A locakratw theorem
constant.
(2) Using the Local-Ratio Corollary, with
r
which is a single edge.
Q.E.D.
4. Putting together NT and the local-ratio theorem Hochbaum [ 12 suggested the following approach to approximate WVC: Let C(V, El, w be the problem’s input, such that w(C*(C))2 1/2 d V ) . (This is achieved by the N T algorithm). Color C by k colors and let I be the “heaviest” monochromatic set of vertices. The cover produced is C= V-I. It follows that
For general graphs she gets the ratio 2 degree) and for planar graphs
-
2 - (A is the maximum
A (k = 4) the performance ration 2 1 . 5 in
timecomplexity of N T and 4-coloring.
We suggest the use of a preparatory algorithm in which all triangles are omitted (with local-ratio 1.5) and therefore, the residual graph is easier to color. Algorithm CO VER 2
Inpu t:
G(V, E), w , k.
Phase 1:
[Triangle elimination]
For every triangle T ( T c V ) do Find 6 = Min { d x ) I x E T ]
For every x E T do w ( x ) + w ( x ) -6 end end c1
f
{x I w ( x ) = 01
36
R Bar- Yehuda and S. Even I/,+
v-c,
Phase 2:
Call NT(G(V1) , w ) to get CO,V o .
Phase 3:
Find a cover approximation, Cz, of G( Vo), by using k-coloring (as in Hochbaum’s approach).
output:
C + c1 u
c, u c,.
2 For k 2 4, R C O V E R ~tc, w ) 1 2 - - , since the local-ratio of Phak se 1 is 1.5 (here I‘ contains of a single graph which is a triangle), the local-ratio of Phase 2 is 1 and the local-ratio of Phase 3 is 2 --2 . k Since the graph colored in Phase 3 is triangle-free, we get the following additional results: 1
( 1 ) For general graphs COVER^(^) I2 --
dii
by using Wigderson’s
approach [ 2 0 ] for coloring a triangle-free graph by k = 2 fl colors in linear time [ 2 11. For planar graphs COVER^(^) I 1.5, and the time-complexity of 4Coloring a triangle-free planar graph is linear. (One uses the fact that in such graphs there is always a vertex of degreel3. See, for example, [ 131. This prevents the need to use a more complex 4Coloring algorithms. Note that the 1.5 performance ratio, for unweighted planar E algorithm” of [SI (or graphs, can be achieved by the “1 [21), however their algorithm is expoexponential time w.r.t. 1/ E and for E = 0.5 is not practical.
+
Before we present our main algorithm we need a few preliminaries. Let the triple (C(V, E), w , k ) (where C is a graph with weight function w and k is a positive integer) be called proper if the following conditions hold: (i)
( 2 k - l ) k 2 I V I.
(ii) There are no odd circuits of length 1 2 k
(iii) d C * ) 2 1 / 2 w(V).
-
1.
37
A locubratio theorem
In the following procedure the statements in square brackets are added for the analysis only, and variablesj (integer), Co, V o , C1, V1,.. (sets), are used only in the brackets.
Procedure CO VER.PR0PER Input:
proper (C(V , E), w , k).
Phase 0:
V ’ V ’ + V, C’+ @ . [ j + 01
Phase 1:
While V ’ Z
C#I
do
Find u E V ’ s . t .w ( v ) = M a x { d u ) 1 u E V’]. Let A o , A l , ...., Ak be the first k
+ 1 layerst
of a Breadth-First-Search (BFSS) on G( V’)starting with
“3
A0 Define B,,
t
t
=uA ,i and B2,
+
I =0
(for j = O , 1,2, ...)
= U A 2i + i =o
f + M i n { s I w ( B ~ ) S ( ~ K - -w~()~. , + ~ 4 . Add Bf to C’.[C’ BfI Remove Bf U B f - from V’. [ Vj Bf U Bf-l, j +
+
+
j+ 11
end output:
C + c:
Proposition 1: Procedure COVER.PROPER satisfies the following properties:
(1) In every application of Phase 1, ~ S K . (2) In every application of Phase 1.Bf-l is an independent set in C ( V %
(3) Ccovers C. f Starting with some m, A,,, A,,,+
$ See for example [7]
...., AK may be empty.
38
(4)
R Bar-Yehudaand S Even
For every iterationj, wfC')'
(1
-
1
1
(1- - * w f V). 2k 1 ( 6 ) RCO VER.PROPER@) 2 -( 5 ) WfC)'
1 - ) . w f Vj). 2k
k '
(7) The time complexity is O( I V I log I V I +I E I).
Proof Assume the contrary. Thus, for every s<_ k , wfB,) > (2k-1). wfBs.,).Thus, [by the assumption] [by (i) of the definition of proper]
[Bo = A 0 =[.}I [V'CVI [by
Y
's definition]
[Bk GV']
An edge between two vertices of A,, s
. [wfVjkwfC,)l.
By summation of the inequality of (4)for every j .
39
A local-ratio theorem
( 6 ) By definition RCOVER.PROPER (G, w ) =
w (C)
wtC*) ( 5 ) above and condition (iii) of properness imply that
. Property
(7) We may start the algorithm by sorting the vertices according to their weights, which requires O(l Vllogl VI) steps. This complexity includes, now, the total time used in Phase 1 for finding M a x [ d u ) Iu E V ’ ) . It is not necessary to continue the BFS of Phase 1, beyond layer f. Thus, each such search is linear in the number of edges t o be eliminated from G(V% and the total time spent in the search of Phase 1 is linear in IEl + IVI. Thus (7) follows.
Q.E.D. Now, the main algorithm.
Algorithm CO VER3 Input:
G(V,E), w
Phase 0:
Find the least integer k s. t. (2k- 1 I k 2 I VI.
Phase 1:
[Elimination of short odd circuits with local-ratio12--].1 k For every odd circuit D c V s.t ID1 5 2 k - 1 d o 6
+
Min [ w ( x ) ~ rE D}
For every x E D do w ( x )
+
w(xJ-6 end
end
c1 +[xlw(x) I/, v-c1.
= 0)
+
Phase 2:
Call N f l C ( V l ) , w l to get Co, V o .
R Bar- Yehuda and S. Even
40
Phase 3:
Call COVER.PROPER(G(Vo), w , k ) to get C2
output:
c+c1 u co u c,.
Proposition 2: Algorithm CO VER3 satisfies the following properties: (1)
1
r C 0 ~ ~ ~ 3 ( n t-< 2
k
.
(2) Its time complexity is the same asNTs. (see Table 2) (3) for unweighted graphs its time complexity is O(l Vl . IEI).
Proof: (1)
The combination of Phase 2 and 3 yields an algorithm with performance ratio 5 2 1 , since the NT algorithm performs localk optimization (ratio= 1) and, for proper graphs, COVER.PROPER 1 has performance ratio 1 2 - - [by Proposition I(6)l. Let be a k simple circuit of length 21-1 thus, iil = 21-1 and TI*= 1; therefore, i) = (21-1)/1 = 2-1/1. Consider, now, the Local-Ratio-Corollary where, I' =[GI I ISk),thus,rr = M a x [ q I I S k ) = 2-l/k and (1) follows.
(2) Let us perform Phase 1 in a slightly different way: Choose a vertex v , and build the first k layers of the BFS starting from v . If
there is an edge u--0, where u and w belong t o the same layer, then an odd-length-circuit D has been detected (see for example [ 151). In this case we find S =Min [ w (x) I x E D , reduce the weights of the vertices in D by 6 and the vertices whose weight is zero are eliminated from the representation of the graph (for the purpose of performing Phase 1). If no such edge (closing an odd circuit) is detected then v is eliminated from the representation of the graph, since no odd-circuit of length I 2 k - 1 passes through it. In any case at least one vertex is eliminated for each BFS. Thus the time complexity is O(l VI . /El). In Phase 2, the best known time-complexity of the N T algorithm
1
A local-ratw theorem
41
(table 2 ) is greater then I VI . IEl.Phase 3 requires O(lEI+I Vllogl Vl), by Proposition l(7). Thus, the whole algorithm is of time complexity as the N T algorithm.
( 3 ) For unweighted graphs, the N T algorithm can be performed in O f I E u m ) time and therefore, the algorithm is of time complexity Of I VI . IEI).
Q.E.D. Corollary: rCO VER3 ( n ) < 2
Proof Define g ( n ) =
-
loglogn 2logn
logn
which is monotone increasing for n 216. loglogn By k-th definition ( 2 k - 3 ) k - 1 < n. Thus, g ( ( 2 l ~ - 3 ) ~)- < ' g(n).
k
We want to show that k < 2S(n). Thus, it suffices to show that ). This is an exercise in elementary mathematics.
< 2g((2k - 3Ik-'
Q.E.D.
5. Extending the local ratio theorem for the weighted set cover problem Let HWVC be the following problem. Given a hypergraph G=(V,E) with weight function w : V+ R+,find a set C c V of minimum total weight s.t. for every e E E, (e n C121. The local-Ratio Theorem and its Corollary hold also for HWVC. Algorithm COVER1 can be applied directly to HWVC with 'COVER 1 < AE [where AE is the maximum edge-degree (or cardinality) in GI. Its running time is linear in the length of the problem's input ( I e I). HWVC is actually the Weighted-Set-Cover Problem' and is an extension of WVC [AE = 21. Even 'Chvatal
[ 4 ] gets, performance-ratio 5
9 f.
i= 1 -
I
i _
[where Ais the maximum vertex degree in GI.
=O(logA).
42
R Bar- Yehuda and S. Even
though we get performance-ratios AE in linear time, we suspect that for any fixed AE, there is no polynomial time approximation algorithm with a better constant performance ratio, unless P=NP [even for the unweighted case], (this is an extension of a conjecture of Hochbaum). 6. Appendix
- NT theorem
Let G(V, El be a simple graph. We denote by C f U ) the subgraph of G induced by U C V and let V’ = { u ’ I u E U].Define the weights of vertices in I/’ by d u ’ ) = d u ) . Nemhauser and Trotter [ 181 presented the following local optiniz ation algorithm:
Algorithm NT
The following theorem states results of Nemhauser and Trotter, but our proof is shorter and does not use linear programming arguments. The sets Co, Vo which Algorithm NT produces, satisfies the following properties:
The NT-Theorem: (i)
If a set D
C_
VOcovers C(Vo)then C = D
U Co covers
G.
(ii) There exists an optimum cover CVGl such that C*(G) 2 Co. (iii) w (C*(G( Vo))) 2 1/2 w ( Vo).
A locakratw theorem
43
Proof Define I . =[x I x $CB ANDx’q! CB]= V-(Vo
U
C,)
Let (x, y ) E E. In order to prove (i) we need to show that either x E Cory E C Case 1 : x E I. , i.e. x. x’ $ CB. Thus, y , Y ’ E C, and therefore Y E co. Case 2: y E I,. Same as Case 1 . Case 3: x E Co or y E C, . This case is trivial. Case 4: x, y E V o . Thus, either x E D or y
E D.
In order to prove (ii), let S = C*(G). Define Sv = S n V,, Sc = S n C,, SI = S n I, and$I= I , Let us show that:
C B ~= ( V - S I )
US>
- Sf
covers B.
(*I
Let (x, y ’ ) E EB.We need to show that eitherx E C B ~ ory’E C B ~ . Case l : x $ S I . T h u s , x E
V-Sl[c_CBII
# S and therefore x $ C, . It follows that y E S [since S covers (x, y ) ] andy E Co [by Case 1, in the proof of (i)]. Thus,y E S n Co [=S,]and thereforey’E S’, [ C C g , 1.
Case 2: x E Sk Thus, x E lo,x
This proves (*). Now, w ( V , ) + 2 4 C , ) = w(V0
u
= UCCB)
c, u C’O)
[by definitions of V O ,CO1
[by (*) and optimality of C B ~
5
(cBl)
=w
((V-S,) u S’C)
= O(Vo u
c, u SI u S’,).
= O(V,) + w ( C , ) + w (S+ +(S,).
It follows that o(C,)
44
R Bar- Yehuda and S. Even
cover of C and contains Co. This proves condition (ii). In order to prove (iii), assume So is an optimum cover of C(V0 ). By (i), Co U So covers G, and by B ’s definition CO U C0 U SO U S’O covers B. Thus,
IW ( C 0 u
C’O
u so u S
[by optimality of CB]
o )
= 2 W ( C O ) +2W(SO).
Therefore, a( Vo) I 2 0 ( S 0 ).
Q.E.D.
References Bar-Yehuda, R., and S. Even, “A Linear Time Approximation Algorithm for the Weighted Vertex Cover Problem”, J. of Algorithms. vol. 2, 1981. 198-203. Bar-Yehuda, R., and S. Even “On Approximating a Vertex Cover for Planar Graphs”, Proc. 14th Ann. ACM Symp. Th. Computing, 1982, 303-309. Bondy, J. A, and U.S.R. Murty, Gruph Theory and Applicarions, North Holland, 1976. Theorem 5.3, page 74. Chvatal, V., “A Greedy-Heuristic for the Set-Covering Problem”, Moth. Operations Res., vol. 4, no. 3, 1979, 233-235. Chiba, S., T. Nishizeki and N., Saito, “Applications of the Lipton and Tarjan’s Planar Separator Theorem”, Jlnfonnotion Proc., 4, 1981, 203-207. Djidjev H. N. “On the Problem of Partitioning Planar graphs”, S A M J on Ak. and Disc. Merh. 3, 2, 1982, 229-240, Even, S., C r q h Algorithms, Comp. Sci. Press, 1979. Garey, M. R., and D. S . Johnson, Computers an Intractability; A Guide to the Theory of NP-Completeness; Freeman, 1979. Cavril, F. see [8]page 134. Galil, 2.. “A new Algorithm for the Maximal Flow Problem”, Proc. 9th Ann. Symp. on Foundations of Computers Science, 1978, 231-245.
A local-ratio theorem
45
Hochbaum, D. S. “Approximation Algorithms for the Set Covering and Vertex Cover Problems, “ S A M J. Cornput.. vol. 1 1 , 3, 1982, 555-556. Hochbaum. D. S., “Efficient Bound3 for the Stable Set, Vertex Cover and Set Packing Problems”, Dis.App. Math. vol. 6 , 1983, 243-254. Harary. F., Graph theory, Addision-Wesly, 1972 (revised). Hopcroft, J . E., and R. M. Karp, “An n5” Algorithm for Maximum Matching in Bipartite Graphs.”SIAM J. Comput., 2, 1973, 225-231. Itai, A,, and M. Rodeh, “Finding a Minimum Circuit in a Graph”, SIAM J. Comput. vol. 7. 4, 1978, 413423. Karp. R. M. “Reducibility among Combinatorial Problems”, in R. E. Miller and J. W. Thatcher (eds.), Complexity of Computer Computations. Plenum Press, New York 85-103. Monien, B. and E. Speckenmeyer, “Some further Approximation Algorithms for the Vertex Cover Problem”, to be presented in the 8th Coll. on Trees in Algebra and Programming. Univ. degli Studi de L’Aquila, Italy. March 1983. Nemhauser, G. L. and L. E. Trotter, Jr., “Vertex Packing: Structural Properties and Algorithms” Mathematical Programming. vol. 8 , 1975, 232-248. Sleator, D. D. K., “An O(mn1ogn) Algorithm for the Maximum Network Flow”, Doctoral Dissertation, Dept. Computer Science, Stanford University, 1980. Wigderson, A., “ h p r o v i n g the Performance Guarantee for Approximate Graph Coloring”, Jatrnal of Assmiorion for Computing Mechinery, vol. 30, 4 , 1983, 729-135.
[ 211
Wigderson, A., private communication.
27
A LOCAL-RATIO THEOREM FOR APPROXIMATING THE WEIGHTED VERTEX COVER PROBLEM
R. BAR-YEHUDA and S. EVEN Computer Science Deportment Technion - Istroel Institute of
Technology
Hoifo, Istroel32000
A local-ratio theorem for approximating the weighted vertex cover problem is presented. It consists of reducing the weights of vertices in certain subgraphs and has the effect of locdaiproximetion. Putting together the Nemhauser-Trotter local optimization algorithm and the localratio theorem yields several new approximation techniques which improve known results from time complexity, simplicity and performanceratio point of view. The main approximation algorithm guarantees a ratio of 2-- 1 where K is the smalloglog?l K lest integer s.t. (2K-1)K2 n (hence: ratio 5 2 1
210gn
t
This is M improvement over the currently known ratios, especially for a “practical” number of vertices (e.g. for graphs which have less than 2400,60000, 1Ol2 vertices the ratio is bounded by 1.75, 1.8, 1.9 respectively).
1 . Introduction A Vertex Cover of a graph is a subset of vertices such that each edge has at least one endpoint in the subset. The Weighted Vertex Cover Problem (WVC) is defined as follows: Given a simple graph G(V, E ) and a weight function o :V + R’, find a cover of minimum total weight. WVC is .known to be NP-Hard, even if all weigts are 1 [ 161 and the graph is planar [81. Therefore, it is natural to look for efficient approximation algorithms.
Let A be an approximation algorithm. For graph G with weight functions w , let C, , C* be the cover A produces and an optimum cover, f All log bases, in this paper, me 2.
28
R Bar- Yehuda and S, Even
respectively . Define R A (G, W ) =
w
fcA)
w
(C*)
and let the performance ratio r A f n ) be
rA(n) = Sup [ R A ~ G w , ) I G = f V, E l where n = I VI] Many approximation algorithms with performance ratio I 2 have been suggested; see, for example Table 1 . No polynomial-time approximation algorithm A is known for which rAfn) 5 2 - c , where E > 0 and fixed. Several approximation algorithms are known for which R A ~ G ,w ) 5 2- EfG), where E depends on G ; e.g. E (C) =- 2
a/ C)
where AfG) is the maximum degree of the vertices of G .
In Section 2 we review the Nemhauser and Trotter [I81 local optimization algorithm f N T ) as well as the observation of Hochbaum [ 1 I I to use it for approximating WVC. In Section 3, a new theorem, the ‘Local-Ratio Theorem’ is presented and proved. It consists of reducing the weights of vertices in certain subgraphs and has the effect of local approximation. As an example of its power we present a trivial proof for the correctness of a linear time approximation algorithm CO VER 1 with ‘COVER l f n ) 1 2 . In Section 4, we present two approximation algorithms in which the NT algorithm and the local-ratio theorem are shown t o be useful.
The first algorithm COVER2 satisfies r ~ 0 v E R 2 f n )<2--for 1
6
ge-
neral graphs, while for planar graphs r c o V E R 2 f n ) < I .5 and its time complexity is O(n2 logn). Hochbaum [121 obtained the same performance ratio, but we manage to avoid the time complexity of 4-coloring. For our last algorithm, co VER3, we prove that rco VER3fn)5 1 - -,1 k where k is the least integer s.t. (2k-1) k 2: n . A similar result, but only for unweighted graphs, has been obtained independently
A local-ratio theorem
29
by Monien and Speckenrneyer [ 17 I.
In Section 5 , the Local-Ratio Theorem is extended t o the Weighted Set-Cover Problem. I
TABLE 1 -SUMMARY OFAPPROXIMATIONRESULTS. (for NT, see Table 2) Performance ratio I
References
Complexity for weighted
Complexity for un weighted
2
2 2 2--
I
E 2
A L
this paper
2--
this paper
2 - loglog v 210g v
fl
“NT”
“NT”
I
“NT”
I
“NT”
“NT”
EV
(For planar graphs)
I
I
I
thispaper
1.6
“NT”
1.5
“NT” -k
1.5
1.5
“NT” -k
“4- COLORING”
“4 -COLORING”
“NT”
“NT”
not applicable 1 +F
* Although this result seems much better then the one shown 2 lines above for unweighted graphs, the algorithm is not useful for practical computations. For example for the akorithm to achieve a ratio I2 one may need ad many as 2 see [2].
160
vertices [ 6 ].For a similar algorithm
R Bar- Yehuda and S. Even
30
G
=(V, E )
Weighted
Unweighted
E2 BV5 B
General
Planar
or EVlog V V2log v
E f l 1.5
2. The Nemhauser and Trotter local optimization algorithm In this short section we review the local optimization algorithm of Nemhauser and Trotter 1181 which is very useful for approximations of wvc t121. Let G(V, El be a simple graph. We denote by G(U) the subgraph of G induced by U c V and let U‘ =( u’lu E U).Define the weights of vertices in U’by d u ’ ) = ~ ( u ) .
The following theorem states results of Nemhauser and Trotter.
The NT-Theorem: The sets C,, Vo which Algorithm NT produces, satisfie the following properties:
A locabratio theorem
(i)
If a set D
31
c V o covers G ( V o ) then C = D U Co covers G.
(ii) There exists an optimum cover C*(G) such that C*(C) 2 Co. [(i) and (ii) are called the local optimally conditions.] (iii) w(C*fCf V o ) ) ) 2 1/2 d V 0 ) . For a proof see [181. An alternate proof is given in the APPENDIX. The significance of the NT-Theorem, as pointed out by Hochbaum 1121, is that the problem of approximating WVC can be limited to graphs G( V o ) which satisfy (iii). A simple illustration of this approach was used by Hochbaum to obtain an algorithm with performace ratio 1 2 : Call NT(C( V h w ) to get Co, V o and return C = Co U Vo . Let us consider now the time-complexity of finding C*(B). which determines the time-complexity on NT.
For the unweighted case the problem can be converted into the maximum matching problem on B (see for example [31 which is of timecomplexity O ( E P ) (see [141). For the weighted case the problem can be converted into a maximum flow problem (see, for example [12]) which is of time-com[I01 or O(EVlogV) 1191. For a summary of the plexity OfE2'3 results, see Table 2.
3. The local-ratio theorem In a previous paper [21 we presented a local approximation technique for the vertex cover problem of unweighted graphs. In this section we present a local approximation technique for the vertex cover problem of weighted graphs. First, we present the following lemma:
Lemma: Let C( V, El be a graph and w , w and o1 be weight functions on V, s.t. for every Y E V: W ( Y ) 2 w , ( v ) + w l ( v ) . Let c*,C: and C: be optimum covers of C with respect t o a,w and a ? . It follows that: w f C * ) 2 w (C*l +02 (Ct).
R Bar- Yehuda and S. Even
32
= O,fC*)+W2fC*)
2 w 1 f C * ) + w 2 f C * ) [by the optimality of Cr and C,*]
Q.E.D.
c
Let be an unweighted graph of fi vertices, whose optimum cover contains C* vertices. Define F=ii/Z*. Let A be an approximation algorithm for WVC and let L O C A L c be the following algorithm:
Algorithm LOCAL Input:
Graph Gf V , E), with weight function w .
Phase I :
Choose a subgraph G f V, E ) of G which is isomorphic to
N
N
hr
c.
Choose 0 565 Min [wfx) Define w o fx) =
I
X E
V] h
wfx)-6
if x E V
wfx)
else.
Phase 2:
Call A f G , w o ) to get Co.
output:
c + c,.
The Local-Ratio Theorem: R L O C A L ~ ~ G w ,) < M a x [F, RA(G, w o ) ] Proof Let c* and c r be the weights of the optimum covers of G with respect to w and w o , and let r = M a x [ i , R A ~ G w, , ) ] , then fC) I
W,
fC)+S.
n
CI
[by I C n V l l n ]
A locabratio theorem
I RA(G, w o ) . co * +i=.6 . C*
33
[by definitions]
[by r's definition]
Ir . ( c o * + 6 P )
< r . c*
[by the lemma]
Q.E.D. Let us consider now a corollary of the Local-Ratio Theorem. Let I' be a finite family of graphs, and rr = MUXIT I C E r}. We denote by G(U), U G V , the subgraph of C(V, E ) induced by U.
Algorithm LOCAL r Input:
G(V, El, w .
Phase 0:
For every x
Phase 1:
For every G( V, some E I', do
- - z), E
V do w (XI + w (XI end subgraph of G which is isomorphic to
6 +Min[wo (X)IXET).
For every x E
7 do w o (x) +-w o(x) -6
end
end
Phase 2:
c1 +[x I w o ( x ) = 01. v1 v-c1 . +
Call A ( G ( V , ), wo) to get Cz.
ou tpu t:
c+c1 u c,.
The Local-Ratio Corollary: R L O C A L ~(C,0 )S Max
[ r r , RA(G(V,), w O ) } .
Proof: By induction on i, the number of iterations of Phase 1, during which 6 > 0.
R Bar- Yehuda and S. Even
34
For i=O the claim is trivial.
+
Suppose the claim holds for i, and for some G, w Phase 1 runs i 1 iterations during which 6 > 0. Let E r be the graph used in the first iteration during which 6 > 0. Observe that we can view the running of LOCAL, as an application of L O C A L c where A (in Phase 2 of LOC A L c ) is replaced by the remaining part of LOCALr. The inductive step is now an immediate consequence of the Local-Ratio Theorem.
Q.E.D. In the applications of LOCALr we shall refer to rr as (a bound on) the local-ratio of Phase 1. Let us demonstrate a simple application of the Local-Ratio Corollary.
Algorithm CO VER 1
Input:
G(V, E), a.
Phase 1:
For every e E E do Let 6 = Min { w ( x E~ e). ~
For every x
Ee
do w (x) = w(x)-6 end
end Output:
c + [ x I d x ) = 01.
This approximation algorithm is essentially the one we described in ill'.
Proposition: For algorithm COVER 1.
(1) The time complexity is O(E)
(2) Its performance ratio 1 2 Proof: (1)
'
The number of operations spent on each edge is bounded by a
I n [I ] w e used a global rather then a local point of vim.
35
A locakratw theorem
constant.
(2) Using the Local-Ratio Corollary, with
r
which is a single edge.
Q.E.D.
4. Putting together NT and the local-ratio theorem Hochbaum [ 12 suggested the following approach to approximate WVC: Let C(V, El, w be the problem’s input, such that w(C*(C))2 1/2 d V ) . (This is achieved by the N T algorithm). Color C by k colors and let I be the “heaviest” monochromatic set of vertices. The cover produced is C= V-I. It follows that
For general graphs she gets the ratio 2 degree) and for planar graphs
-
2 - (A is the maximum
A (k = 4) the performance ration 2 1 . 5 in
timecomplexity of N T and 4-coloring.
We suggest the use of a preparatory algorithm in which all triangles are omitted (with local-ratio 1.5) and therefore, the residual graph is easier to color. Algorithm CO VER 2
Inpu t:
G(V, E), w , k.
Phase 1:
[Triangle elimination]
For every triangle T ( T c V ) do Find 6 = Min { d x ) I x E T ]
For every x E T do w ( x ) + w ( x ) -6 end end c1
f
{x I w ( x ) = 01
36
R Bar- Yehuda and S. Even I/,+
v-c,
Phase 2:
Call NT(G(V1) , w ) to get CO,V o .
Phase 3:
Find a cover approximation, Cz, of G( Vo), by using k-coloring (as in Hochbaum’s approach).
output:
C + c1 u
c, u c,.
2 For k 2 4, R C O V E R ~tc, w ) 1 2 - - , since the local-ratio of Phak se 1 is 1.5 (here I‘ contains of a single graph which is a triangle), the local-ratio of Phase 2 is 1 and the local-ratio of Phase 3 is 2 --2 . k Since the graph colored in Phase 3 is triangle-free, we get the following additional results: 1
( 1 ) For general graphs COVER^(^) I2 --
dii
by using Wigderson’s
approach [ 2 0 ] for coloring a triangle-free graph by k = 2 fl colors in linear time [ 2 11. For planar graphs COVER^(^) I 1.5, and the time-complexity of 4Coloring a triangle-free planar graph is linear. (One uses the fact that in such graphs there is always a vertex of degreel3. See, for example, [ 131. This prevents the need to use a more complex 4Coloring algorithms. Note that the 1.5 performance ratio, for unweighted planar E algorithm” of [SI (or graphs, can be achieved by the “1 [21), however their algorithm is expoexponential time w.r.t. 1/ E and for E = 0.5 is not practical.
+
Before we present our main algorithm we need a few preliminaries. Let the triple (C(V, E), w , k ) (where C is a graph with weight function w and k is a positive integer) be called proper if the following conditions hold: (i)
( 2 k - l ) k 2 I V I.
(ii) There are no odd circuits of length 1 2 k
(iii) d C * ) 2 1 / 2 w(V).
-
1.
37
A locubratio theorem
In the following procedure the statements in square brackets are added for the analysis only, and variablesj (integer), Co, V o , C1, V1,.. (sets), are used only in the brackets.
Procedure CO VER.PR0PER Input:
proper (C(V , E), w , k).
Phase 0:
V ’ V ’ + V, C’+ @ . [ j + 01
Phase 1:
While V ’ Z
C#I
do
Find u E V ’ s . t .w ( v ) = M a x { d u ) 1 u E V’]. Let A o , A l , ...., Ak be the first k
+ 1 layerst
of a Breadth-First-Search (BFSS) on G( V’)starting with
“3
A0 Define B,,
t
t
=uA ,i and B2,
+
I =0
(for j = O , 1,2, ...)
= U A 2i + i =o
f + M i n { s I w ( B ~ ) S ( ~ K - -w~()~. , + ~ 4 . Add Bf to C’.[C’ BfI Remove Bf U B f - from V’. [ Vj Bf U Bf-l, j +
+
+
j+ 11
end output:
C + c:
Proposition 1: Procedure COVER.PROPER satisfies the following properties:
(1) In every application of Phase 1, ~ S K . (2) In every application of Phase 1.Bf-l is an independent set in C ( V %
(3) Ccovers C. f Starting with some m, A,,, A,,,+
$ See for example [7]
...., AK may be empty.
38
(4)
R Bar-Yehudaand S Even
For every iterationj, wfC')'
(1
-
1
1
(1- - * w f V). 2k 1 ( 6 ) RCO VER.PROPER@) 2 -( 5 ) WfC)'
1 - ) . w f Vj). 2k
k '
(7) The time complexity is O( I V I log I V I +I E I).
Proof Assume the contrary. Thus, for every s<_ k , wfB,) > (2k-1). wfBs.,).Thus, [by the assumption] [by (i) of the definition of proper]
[Bo = A 0 =[.}I [V'CVI [by
Y
's definition]
[Bk GV']
An edge between two vertices of A,, s
. [wfVjkwfC,)l.
By summation of the inequality of (4)for every j .
39
A local-ratio theorem
( 6 ) By definition RCOVER.PROPER (G, w ) =
w (C)
wtC*) ( 5 ) above and condition (iii) of properness imply that
. Property
(7) We may start the algorithm by sorting the vertices according to their weights, which requires O(l Vllogl VI) steps. This complexity includes, now, the total time used in Phase 1 for finding M a x [ d u ) Iu E V ’ ) . It is not necessary to continue the BFS of Phase 1, beyond layer f. Thus, each such search is linear in the number of edges t o be eliminated from G(V% and the total time spent in the search of Phase 1 is linear in IEl + IVI. Thus (7) follows.
Q.E.D. Now, the main algorithm.
Algorithm CO VER3 Input:
G(V,E), w
Phase 0:
Find the least integer k s. t. (2k- 1 I k 2 I VI.
Phase 1:
[Elimination of short odd circuits with local-ratio12--].1 k For every odd circuit D c V s.t ID1 5 2 k - 1 d o 6
+
Min [ w ( x ) ~ rE D}
For every x E D do w ( x )
+
w(xJ-6 end
end
c1 +[xlw(x) I/, v-c1.
= 0)
+
Phase 2:
Call N f l C ( V l ) , w l to get Co, V o .
R Bar- Yehuda and S. Even
40
Phase 3:
Call COVER.PROPER(G(Vo), w , k ) to get C2
output:
c+c1 u co u c,.
Proposition 2: Algorithm CO VER3 satisfies the following properties: (1)
1
r C 0 ~ ~ ~ 3 ( n t-< 2
k
.
(2) Its time complexity is the same asNTs. (see Table 2) (3) for unweighted graphs its time complexity is O(l Vl . IEI).
Proof: (1)
The combination of Phase 2 and 3 yields an algorithm with performance ratio 5 2 1 , since the NT algorithm performs localk optimization (ratio= 1) and, for proper graphs, COVER.PROPER 1 has performance ratio 1 2 - - [by Proposition I(6)l. Let be a k simple circuit of length 21-1 thus, iil = 21-1 and TI*= 1; therefore, i) = (21-1)/1 = 2-1/1. Consider, now, the Local-Ratio-Corollary where, I' =[GI I ISk),thus,rr = M a x [ q I I S k ) = 2-l/k and (1) follows.
(2) Let us perform Phase 1 in a slightly different way: Choose a vertex v , and build the first k layers of the BFS starting from v . If
there is an edge u--0, where u and w belong t o the same layer, then an odd-length-circuit D has been detected (see for example [ 151). In this case we find S =Min [ w (x) I x E D , reduce the weights of the vertices in D by 6 and the vertices whose weight is zero are eliminated from the representation of the graph (for the purpose of performing Phase 1). If no such edge (closing an odd circuit) is detected then v is eliminated from the representation of the graph, since no odd-circuit of length I 2 k - 1 passes through it. In any case at least one vertex is eliminated for each BFS. Thus the time complexity is O(l VI . /El). In Phase 2, the best known time-complexity of the N T algorithm
1
A local-ratw theorem
41
(table 2 ) is greater then I VI . IEl.Phase 3 requires O(lEI+I Vllogl Vl), by Proposition l(7). Thus, the whole algorithm is of time complexity as the N T algorithm.
( 3 ) For unweighted graphs, the N T algorithm can be performed in O f I E u m ) time and therefore, the algorithm is of time complexity Of I VI . IEI).
Q.E.D. Corollary: rCO VER3 ( n ) < 2
Proof Define g ( n ) =
-
loglogn 2logn
logn
which is monotone increasing for n 216. loglogn By k-th definition ( 2 k - 3 ) k - 1 < n. Thus, g ( ( 2 l ~ - 3 ) ~)- < ' g(n).
k
We want to show that k < 2S(n). Thus, it suffices to show that ). This is an exercise in elementary mathematics.
< 2g((2k - 3Ik-'
Q.E.D.
5. Extending the local ratio theorem for the weighted set cover problem Let HWVC be the following problem. Given a hypergraph G=(V,E) with weight function w : V+ R+,find a set C c V of minimum total weight s.t. for every e E E, (e n C121. The local-Ratio Theorem and its Corollary hold also for HWVC. Algorithm COVER1 can be applied directly to HWVC with 'COVER 1 < AE [where AE is the maximum edge-degree (or cardinality) in GI. Its running time is linear in the length of the problem's input ( I e I). HWVC is actually the Weighted-Set-Cover Problem' and is an extension of WVC [AE = 21. Even 'Chvatal
[ 4 ] gets, performance-ratio 5
9 f.
i= 1 -
I
i _
[where Ais the maximum vertex degree in GI.
=O(logA).
42
R Bar- Yehuda and S. Even
though we get performance-ratios AE in linear time, we suspect that for any fixed AE, there is no polynomial time approximation algorithm with a better constant performance ratio, unless P=NP [even for the unweighted case], (this is an extension of a conjecture of Hochbaum). 6. Appendix
- NT theorem
Let G(V, El be a simple graph. We denote by C f U ) the subgraph of G induced by U C V and let V’ = { u ’ I u E U].Define the weights of vertices in I/’ by d u ’ ) = d u ) . Nemhauser and Trotter [ 181 presented the following local optiniz ation algorithm:
Algorithm NT
The following theorem states results of Nemhauser and Trotter, but our proof is shorter and does not use linear programming arguments. The sets Co, Vo which Algorithm NT produces, satisfies the following properties:
The NT-Theorem: (i)
If a set D
C_
VOcovers C(Vo)then C = D
U Co covers
G.
(ii) There exists an optimum cover CVGl such that C*(G) 2 Co. (iii) w (C*(G( Vo))) 2 1/2 w ( Vo).
A locakratw theorem
43
Proof Define I . =[x I x $CB ANDx’q! CB]= V-(Vo
U
C,)
Let (x, y ) E E. In order to prove (i) we need to show that either x E Cory E C Case 1 : x E I. , i.e. x. x’ $ CB. Thus, y , Y ’ E C, and therefore Y E co. Case 2: y E I,. Same as Case 1 . Case 3: x E Co or y E C, . This case is trivial. Case 4: x, y E V o . Thus, either x E D or y
E D.
In order to prove (ii), let S = C*(G). Define Sv = S n V,, Sc = S n C,, SI = S n I, and$I= I , Let us show that:
C B ~= ( V - S I )
US>
- Sf
covers B.
(*I
Let (x, y ’ ) E EB.We need to show that eitherx E C B ~ ory’E C B ~ . Case l : x $ S I . T h u s , x E
V-Sl[c_CBII
# S and therefore x $ C, . It follows that y E S [since S covers (x, y ) ] andy E Co [by Case 1, in the proof of (i)]. Thus,y E S n Co [=S,]and thereforey’E S’, [ C C g , 1.
Case 2: x E Sk Thus, x E lo,x
This proves (*). Now, w ( V , ) + 2 4 C , ) = w(V0
u
= UCCB)
c, u C’O)
[by definitions of V O ,CO1
[by (*) and optimality of C B ~
5
(cBl)
=w
((V-S,) u S’C)
= O(Vo u
c, u SI u S’,).
= O(V,) + w ( C , ) + w (S+ +(S,).
It follows that o(C,)
44
R Bar- Yehuda and S. Even
cover of C and contains Co. This proves condition (ii). In order to prove (iii), assume So is an optimum cover of C(V0 ). By (i), Co U So covers G, and by B ’s definition CO U C0 U SO U S’O covers B. Thus,
IW ( C 0 u
C’O
u so u S
[by optimality of CB]
o )
= 2 W ( C O ) +2W(SO).
Therefore, a( Vo) I 2 0 ( S 0 ).
Q.E.D.
References Bar-Yehuda, R., and S. Even, “A Linear Time Approximation Algorithm for the Weighted Vertex Cover Problem”, J. of Algorithms. vol. 2, 1981. 198-203. Bar-Yehuda, R., and S. Even “On Approximating a Vertex Cover for Planar Graphs”, Proc. 14th Ann. ACM Symp. Th. Computing, 1982, 303-309. Bondy, J. A, and U.S.R. Murty, Gruph Theory and Applicarions, North Holland, 1976. Theorem 5.3, page 74. Chvatal, V., “A Greedy-Heuristic for the Set-Covering Problem”, Moth. Operations Res., vol. 4, no. 3, 1979, 233-235. Chiba, S., T. Nishizeki and N., Saito, “Applications of the Lipton and Tarjan’s Planar Separator Theorem”, Jlnfonnotion Proc., 4, 1981, 203-207. Djidjev H. N. “On the Problem of Partitioning Planar graphs”, S A M J on Ak. and Disc. Merh. 3, 2, 1982, 229-240, Even, S., C r q h Algorithms, Comp. Sci. Press, 1979. Garey, M. R., and D. S . Johnson, Computers an Intractability; A Guide to the Theory of NP-Completeness; Freeman, 1979. Cavril, F. see [8]page 134. Galil, 2.. “A new Algorithm for the Maximal Flow Problem”, Proc. 9th Ann. Symp. on Foundations of Computers Science, 1978, 231-245.
A local-ratio theorem
45
Hochbaum, D. S. “Approximation Algorithms for the Set Covering and Vertex Cover Problems, “ S A M J. Cornput.. vol. 1 1 , 3, 1982, 555-556. Hochbaum. D. S., “Efficient Bound3 for the Stable Set, Vertex Cover and Set Packing Problems”, Dis.App. Math. vol. 6 , 1983, 243-254. Harary. F., Graph theory, Addision-Wesly, 1972 (revised). Hopcroft, J . E., and R. M. Karp, “An n5” Algorithm for Maximum Matching in Bipartite Graphs.”SIAM J. Comput., 2, 1973, 225-231. Itai, A,, and M. Rodeh, “Finding a Minimum Circuit in a Graph”, SIAM J. Comput. vol. 7. 4, 1978, 413423. Karp. R. M. “Reducibility among Combinatorial Problems”, in R. E. Miller and J. W. Thatcher (eds.), Complexity of Computer Computations. Plenum Press, New York 85-103. Monien, B. and E. Speckenmeyer, “Some further Approximation Algorithms for the Vertex Cover Problem”, to be presented in the 8th Coll. on Trees in Algebra and Programming. Univ. degli Studi de L’Aquila, Italy. March 1983. Nemhauser, G. L. and L. E. Trotter, Jr., “Vertex Packing: Structural Properties and Algorithms” Mathematical Programming. vol. 8 , 1975, 232-248. Sleator, D. D. K., “An O(mn1ogn) Algorithm for the Maximum Network Flow”, Doctoral Dissertation, Dept. Computer Science, Stanford University, 1980. Wigderson, A., “ h p r o v i n g the Performance Guarantee for Approximate Graph Coloring”, Jatrnal of Assmiorion for Computing Mechinery, vol. 30, 4 , 1983, 729-135.
[ 211
Wigderson, A., private communication.