- Email: [email protected]
Approximating Node-Deletion Problems for Matroidal Properties
Journal of Algorithms 31, 211᎐227 Ž1999. Article ID jagm.1998.0995, available online at http:rrwww.idealibrary.com on
Approximating Node-Deletion Problems for Matroidal Properties* Toshihiro Fujito Department of Electrical Engineering, Hiroshima Uni¨ ersity, 1-4-1 Kagamiyama, Higashi-Hiroshima 739 Japan E-mail: [email protected] Received October 22, 1997; revised November 2, 1998
The paper is concerned with the polynomial time approximability of node-deletion problems for hereditary properties. It is observed that, when such a property derives a matroid on any graph, the problem can be formulated as a matroid set cover optimization problem, and this leads us naturally to consider two well-known approaches, greedy and primal-dual. The heuristics based on these principles are analyzed and general approximation bounds are obtained. Next, more specific types of hereditary properties, called uniformly sparse, are introduced and, for any of them, the primal-dual heuristic is shown to approximate the corresponding nodedeletion problem within a factor of 2. We conclude that there exist infinitely many Žat least countably many. node-deletion problems, each of which approximable to a factor of 2, for hereditary properties with an infinite number of minimal forbidden graphs. 䊚 1999 Academic Press
1. INTRODUCTION The paper is concerned with the polynomial time approximability of node-deletion problems for hereditary properties. The node-deletion problem for a graph property Ždenoted NDŽ . throughout the paper. is a typical graph optimization problem; that is, give a node-weighted graph G, find a vertex set of the minimum weight sum whose deletion Žalong with all the incident edges. from G leaves a subgraph satisfying the property . A graph property is hereditary if every subgraph of a graph satisfying * A preliminary version presented at the 24th International Colloquium on Automata, Languages and Programming, Bologna, Italy, July 1997. 211 0196-6774r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
212
TOSHIHIRO FUJITO
also satisfies . A number of well-studied graph properties are hereditary such as independent set, planar, bipartite, acyclic, degree-constrained, circular-arc, circle graph, chordal, comparability, permutation, perfect. Naturally, many well-known graph problems fall into this class of problems when desired graph properties are specified appropriately. Lewis and Yannakakis proved, however, that whenever is nontrivial and hereditary on induced subgraphs NDŽ . is NP-hard w13x. When this general result of NP-hardness was established, almost nothing was known about the polynomial time approximability of NDŽ .’s except for good approximation algorithms for the vertex cover ŽVC. problem. Moreover, their generic reductions from VC to other NDŽ .’s are approximation preserving, and as such, no NDŽ . can be approximated better than VC can be. Thus it was natural for them to pose a question: can other node-deletion problems be approximated well? It has been long known that VC can be approximated with ratio 2 Žachievable by a simple maximal matching based heuristic w10, p. 134x for unweighted graphs. and a better approximation had been a subject of extensive research over the years. Yet the best constant bound has remained the same at 2 while the best known heuristics can accomplish only slightly better Ž2 y log log nr2 log n of w3, 15x.. On the other hand, very few other NDŽ .’s are known to be approximable within some constant factor even today, not to mention a factor of 2. As observed in w14x whenever hereditary has only a finite number of minimal forbidden graphs NDŽ . can be efficiently approximated within some constant factor of the optimum. It was in fact conjectured therein that these are the only node-deletion problems approximable within some constant factor Žsee also w19x.. It was found later, however, that this conjecture does not hold, as the Žunweighted. feedback vertex set ŽFVS. problem Ži.e., s‘‘acyclic’’. on undirected graphs was shown to be approximable within a factor of 4 w4x Žnote, a simple cycle of every length is a minimal forbidden graph for this .. While even the weighted version was later found to be constantly approximable Žsee below., until now this problem has been the only known exception to the conjecture of Lund and Yannakakis. 1.1. Our Results We will show that there exist infinitely many Žat least countably many. constant factor approximable NDŽ .’s for with an infinite number of minimal forbidden graphs. For that purpose we shall concentrate on such hereditary properties that derive matroids on Žthe edge set of. any graph Ždetails given later.. The class of NDŽ .’s for such properties includes VC, FVS, and many others.
APPROXIMATING NODE-DELETION PROBLEMS
213
The problems for such ’s will be cast in a form of matroid set cover optimization problems. This reformulation leads us naturally to consider two well-known approaches, greedy and primal-dual. The heuristic based on each principle is analyzed and general approximation bounds are obtained for NDŽ .’s. It turns out that our primal-dual algorithm is simpler than those primal-dual based 2-approximation algorithms designed particularly for FVS Žsee below.. Next, to capture a class of constant factor approximable NDŽ .’s, more specific types of hereditary properties are introduced. It will be shown that, for such properties called ‘‘uniformly sparse,’’ a certain combinatorial bound holds, in terms of matroid ranks, for any minimal solution of NDŽ .. This allows us to apply the general bound of primal-dual for this class of NDŽ .’s and conclude that, for any of Žinfinitely many. uniformly sparse properties Žeach having an infinite number of minimal forbidden graphs., NDŽ . is approximable within a factor of 2, matching the best constant factor known for either VC or FVS. 1.2. Other Related Work Every NDŽ . for nontrivial hereditary is MAX SNP-hard, as pointed out in w14x, due to the reductions of w13x and the result of w16x. Thus, there exists some positive constant ⑀ such that no polynomial time algorithm for NDŽ . can achieve an approximation factor better than 1 q ⑀ , unless P s NP w1x. Yet a better lower bound is provided by the one in approximation of VC as it serves as a lower bound for every NDŽ . for hereditary . The lower bound for VC approximation has been continuously improved in the last few years and currently it is known to be arbitrarily close to 7 Ž . w12x. 6 modulo P / NP The approximation ratio of 4 for the unweighted FVS w4x was subsequently extended to the one for the weighted FVS and was further improved to 2 w2, 5x. Recently Chudak et al. gave a primal-dual interpretation of these 2-approximation algorithms w6x and also presented a new primal-dual algorithm for FVS which has the same performance ratio but is slightly simpler than the previous two. All of these 2-approximation algorithms as well as ours, based on the same principle, are similar in their execution, repeatedly reducing a certain form of weight distributions defined on a certain set of subgraphs. Our primal-dual algorithm is simpler than others, however, since it does not look into nor modify explicitly any special structure in graphs under consideration. Some nontrivial Žalbeit nonconstant. approximation factors known for other specific NDŽ .’s include O Žlog < V <. for s‘‘bipartite’’ w11x and O Žlog < V
214
TOSHIHIRO FUJITO
2. PRELIMINARIES 2.1. Notation and Definitions A subgraph of G s Ž V, E . induced by X : V is denoted by Gw X x. For X : V let Ew X x denote the set of edges induced by X, and conversely, let V w F x for F : E denote the set of vertices incident to some edge in F. Ew X, Y x is the set of edges with one end in X and the other in Y. For any graph G let V Ž G . and EŽ G . denote the vertex set and the edge set of G, respectively. The set of edges incident to some vertex of X is denoted ␦ Ž X . and when those edges are restricted to the ones in a subgraph Gw Y x we denote it by ␦ Y Ž X .Žs ␦ Ž X . l Ew Y x.. Let ␦ Ž u. Ž ␦ Y Ž u., respectively. be a shorthand of ␦ Ž u4. Ž ␦ Y Ž u4., respectively.. A graph property is nontri¨ ial if infinitely many graphs satisfy and infinitely many graphs fail to satisfy it. It is hereditary Ž on induced subgraphs. if, in any graph satisfying , every Žresp., vertex-induced. subgraph also satisfies . For a hereditary property any graph which does not satisfy is called a forbidden graph for , and it is a minimal one if, additionally, every ‘‘proper’’ Žresp., induced. subgraph of it satisfies . Any hereditary property is equivalently characterized by the set of all the minimal forbidden graphs for . It is customary to measure the quality of an approximation algorithm by its performance ratio, which is the worst case ratio of the optimal value to the value of an approximate solution returned by the algorithm for the same instance. 2.2. Matroidal Properties One way to represent a matroid M is by a pair Ž E, r ., where E is the ground set and r is the rank function defined on 2 E, of M. A set F : E is called ᎏindependent if r Ž F . s < F < Žand conversely, r Ž F⬘. is the cardinality of a largest independent subset of F⬘ for an arbitrary F⬘ : E ., ᎏdependent if r Ž F . - < F <, ᎏa base if it is a maximal Žand, hence, maximum in any matroid. independent set, ᎏa circuit if it is a minimal dependent set, ᎏspanning if r Ž F . s r Ž E .. For any matroid M s Ž E, r . there is the dual matroid M d s Ž E, r d . defined on the same ground set E. The rank functions r and r d are related such that 䢇
䢇
r dŽ E y F . s Ž < E< y r Ž E. . y Ž < F < y r Ž F . . for any F : E.
APPROXIMATING NODE-DELETION PROBLEMS
215
For two matroids M1 s Ž E1 , r 1 . and M2 s Ž E2 , r 2 . defined on disjoint sets E1 and E2 , the direct sum of M1 and M2 , written M1 [ M2 , is the matroid on E1 j E2 such that its rank function r is given by r Ž F . s r 1Ž F l E1 . q r 2 Ž F l E2 . for any subset F : E1 j E2 . It is easy to verify that the dual of the direct sum of M1 and M2 is the direct sum of the duals of M1 and M2 ; that is, Ž M1 [ M2 . d s M1d [ M2d. If I Ž M . is the set of independent sets of matroid M on E, I: I g I Ž M ., I : F 4 for any F : E is also the set of independent sets of a matroid defined on F. Such a matroid is called a restriction of M to F; the rank function here is just the rank function of M but is restricted to subsets of F. ŽFor more on matroid theory see, e.g., w17x.. 䢇
䢇
We say that a hereditary graph property is matroidal if, on any graph G s Ž V, E ., the edge sets of subgraphs of G satisfying form the set of independent sets of a matroid defined on E. Such a matroid is said to be deri¨ ed from the property on E. The property s‘‘acyclic’’ is a typical example of such properties, where the family of edge sets of acyclic subgraphs defines a cycle matroid on any graph. 䢇
Denote by I Ž G . the family of subgraphs of G satisfying and consider any subgraph H of G. Then, for any , I Ž H . is I Ž G . restricted to H, i.e., I Ž G . l G⬘: G⬘ is a subgraph of H 4 . Therefore, if is matroidal, the matroid derived from on EŽ H . is necessarily a restriction of the one derived from on EŽ G .. This means that the rank functions of matroids derived from any are identical, differing only in their domains. For this reason the rank function of a derived matroid on any Žedge set of. graph will be denoted by r Žalong with specification of its domain.. Note also that, by definition, when a matroid M is derived from on EŽ G ., a subgraph H of G is a minimal forbidden graph for iff EŽ H . is a circuit of M. 䢇
A set function f : 2 S ª ⺢ is submodular if for any subsets A, B
of S: f Ž A. q f Ž B . G f Ž A l B . q f Ž A j B . .
3. APPROXIMATION ALGORITHMS FOR MATROIDAL PROPERTIES 3.1. Formulations Perhaps a most natural integer program formulation of NDŽ ., when is general hereditary, is the following ‘‘hitting set’’ formulation: using a 0᎐1 vector x as an incidence vector to represent a feasible vertex set
216
TOSHIHIRO FUJITO
F : V, it looks like Min
Ý
wu x u
ugV
subject to:
Ý
x u G 1,
H g ⍀G Ž . ,
Ž IP1 .
ugV Ž H .
x u g 0, 1 4 ,
u g V,
where ⍀ G Ž . is the set of minimal forbidden graphs for contained as subgraphs in G. It was tried by Bar-Yehuda et al. to approximate the weighted FVS using a primal-dual strategy based on this IP for FVS and its LP dual, which resulted in a factor O Žlog < V <. approximation w4x. Moreover, it was indicated in w9x that in case of the FVS problem the linear relaxation of the formulation above introduces the integrality gap Ži.e, the ratio between the integer and fractional optima. of size as large as ⍀ Žlog < V <.. While other IP formulations were found to bound the integrality gap by a factor of 2 for the FVS problem Žsee w6x., we shall show later that there exists another formulation with integrality gap of at most 2 for many NDŽ .’s, including FVS Žsee Corollary 8.. Let us assume henceforth that is matroidal. Taking advantage of the special structure of solution sets for such NDŽ .’s, we reformulate it in the form of a matroid optimization problem, where, instead of vertices, edge sets of form ␦ Ž u. are used to represent basic constituents of an instance: Matroid formulation. Given a weighted set system Ž E, EV s ␦ Ž u.: u g V 4 , w: EV ª ⺡q. and the matroid M s Ž E, r . derived from , find EF s a subcollection EF : EV of the minimum weight sum such that jE Dug F ␦ Ž u. s ␦ Ž F . is spanning in M d. Or, more concisely, min F:V
½ Ý w : r Ž ␦ Ž F . . s r Ž E. 5 . d
d
u
ugF
The equivalence follows easily from the fact that F : V is a solution of NDŽ . iff E y ␦ Ž F . is independent in M s Ž E, r . Žwhich is equivalent to say ␦ Ž F . is spanning in M d .. Now, based on this interpretation of NDŽ ., we come up with a new IP formulation, reflecting the matroid formulation. The question here is how to replace the constraint r d Ž ␦ Ž F .. s r d ŽDug F ␦ Ž u.. s r d Ž E . by linear inequalities in x. Our solution given below is to have a linear inequality of form Ý ug S l F r d Ž ␦S Ž u.. G r d Ž Ew S x. which is valid at Gw S x for every induced subgraph Gw S x.
APPROXIMATING NODE-DELETION PROBLEMS
217
We claim that NDŽ . on graph G s Ž V, E . can be formulated by the following IP when M s Ž E, r . is the matroid derived on G from : Min
Ý
wu x u
ugV
subject to:
Ý r d Ž ␦S Ž u . . x u G r d Ž E w S x . ,
S : V,
Ž IP2 .
ugS
x u g 0, 1 4 ,
u g V.
THEOREM 1. When M s Ž E, r . is the matroid deri¨ ed on G from , F is a solution of NDŽ . iff x F g 0, 14V Ž incidence ¨ ector of F . is a feasible solution to Ž IP2 .. Proof. ŽOnly if. F : V is a solution in G m F l S is a solution in G w S x
᭙S : V
m ␦S Ž FlS . is spanning in M d Ž G w S x . m r d Ž ␦S Ž F l S . . s r d Ž E w S x . d
Because r is a nonnegative submodular function Ý i r any family of Ei ’s, and hence, ᭙S : V,
Ý r d Ž ␦S Ž u . . x uF s Ý ugS
dŽ
᭙S:V
᭙S : V .
Ei . G r d ŽDi Ei . for
r d Ž ␦S Ž u . . G r d Ž ␦S Ž F l S . . s r d Ž E w S x . .
ugFlS
ŽIf. Suppose F is not a solution of NDŽ .. This implies the existence of a minimal forbidden graph H for , as a subgraph of G, avoided by F. Since F l V Ž H . s ⭋,
Ý
r d Ž ␦ V Ž H . Ž u . . x uF s 0.
ugV Ž H .
Since EŽ H . is a circuit of M Ž G . Žand of M Ž H .., V Ž H . induces an edge set which is dependent in M Ž G . Žand in M Ž H ... This implies r Ž Ew V Ž H .x. < Ew V Ž H .x<, and hence, rdŽ E VŽ H .
.s
E VŽ H .
y rŽ E VŽ H .
. ) 0.
Thus, x F does not satisfy the constraint corresponding to V Ž H .. 3.2. Greedy Heuristic Given a Žmatroid. set system based formulation and its resemblance to the standard set cover problem, the greedy heuristic is an approach to come in due course. In the current context such a heuristic, called
218
TOSHIHIRO FUJITO
FIG. 1. Greedy algorithm Greedy for NDŽ ..
Greedy, repeatedly selects a subset Žof form ␦ Ž u.. for inclusion, which gives a maximal increase in the rank of a current solution per weight Žsee Fig. 1.. Wolsey showed that the well-known analysis of Chvatal ´ w7x for greedy set cover approximation generalizes to the case of a matroid set system w18x Žactually, even to more general submodular systems.; that is, PROPOSITION 2. Let H Ž i . denote the ith Harmonic number, and ⌬ s max ug V r d Ž ␦ Ž u... On any graph G s Ž V, E . Greedy finds a solution of NDŽ . with its weight bounded by H Ž ⌬ . F 1 q ln ⌬ times the optimum. def
Thus, Greedy approximates NDŽ . within an O Žlog n. factor for any matroidal Žinterestingly, as mentioned earlier, the ŽIP1 .-based primaldual heuristic for FVS achieves a tight performance ratio of the same order.. One serious drawback of Greedy, however, is that it cannot deliver any constant factor approximation in general, even for the simplest case of VC problem. 3.3. Primal-Dual Heuristic We now turn to an alternative approach, primal-dual, and we do so using the IP formulation ŽIP2 .. Take first the dual of the LP relaxation of ŽIP2 . Žin which every integrality constraint on x u is replaced by x u G 0.: Max
Ý r d Ž E w S x . yS S:V
subject to:
Ý
r d Ž ␦S Ž u . . yS F wu ,
u g V,
Ž D2 .
S: ugS
yS G 0,
S : V.
The primal-dual approximation algorithm PD, based on ŽIP2 . and ŽD 2 . is presented in Fig. 2. We elaborate more on it. PD starts with F s ⭋, the original graph Gw S⬘x s Ž V, E . and the dual feasible solution y s 0. Given F, if it is not yet a solution of NDŽ . there must exist some set S : V
APPROXIMATING NODE-DELETION PROBLEMS
219
FIG. 2. Primal-Dual Algorithm PD for NDŽ ..
corresponding to a violated constraint of ŽIP2 .. In particular the set of all the remaining vertices S⬘Žs V y F . must be always such a set, and thus we can always choose S⬘ as a ‘‘violated set.’’ PD then increases the dual variable yS⬘ as much as possible until for some vertex u in S⬘ the dual constraint for u becomes tight; i.e., r d Ž ␦S Ž u . . yS s wu .
Ý
Ž 1.
S: ugS
Notice that yS⬘ here can be increased indeed because S⬘ is the collection of all those vertices whose corresponding dual constraints were not yet tight. PD adds u into a solution set F and at the same time removes it from S⬘. Clearly F eventually becomes a solution of NDŽ . Žand to ŽIP2 .. while y is kept feasible to ŽD 2 .. Lastly, the vertices in F are examined one by one, in th reverse order of their inclusion to F, and whenever any of them is found to be extraneous it is thrown out of F. PD clearly constructs a feasible solution F of NDŽ . and a solution y feasible for ŽD 2 .. These two solution are related, by means of Ž1., such that
Ý ugF
wu s
r d Ž ␦S Ž u . . ys s
Ý Ý ugF S : ugS
Ý S:V
ž
Ý ugSlF
r d Ž ␦S Ž u . . yS .
/
Ž 2.
The analysis of the performance ratio of PD can be reduced to that of the following combinatorial bound. THEOREM 3. Let M s Ž EŽ G ., r . be the matroid deri¨ ed from . Then the performance ratio of the primal-dual algorithm PD is bounded by max
½
Ý ug X r d Ž ␦ Ž u . . r d Ž EŽ G. .
5
,
where max is taken o¨ er any minimal solution X of NDŽ . in any G.
220
TOSHIHIRO FUJITO
Proof. Let F denote the output solution of PD. It follows from Ž2. Žand the form of the objective function in ŽD 2 .. that the approximation ratio is reduced to the maximum ratio between Ý ug S l F r d Ž ␦S Ž u.. and r d Ž Ew S x. for any S with nonzero yS . Let Sl be S⬘ chosen by the lth iteration of the While-loop; that is, Sl s V y u1 , . . . , u ly14 , and they are the only S’s with yS ) 0. Thus, it suffices to show that Sl l F is a minimal solution in Gw Sl x for all l G 1. Recall that the last clean-up step of the algorithm examines, in the decreasing order of l, if u l ’s are needed for F to be a solution in G. Let Fj b F remaining after all u l ’s, for l G j, are examined by this For-loop. Then, S j l F s S j l Fj for all j G 1. Assume inductively that Sl l Fl is a minimal solution in Gw Sl x for all l G j q 1. Now if u j f Fj it is clear that Fl l Sl is a minimal solution in Gw Sl x for all l G j. Otherwise, u j g Fj because Fj y u j 4 is not a solution in G, but it implies that neither is Ž S j l Fj . y u j 4 in Gw S j x. Therefore, S j l Fj Žs S j l F . must be a minimal solution in Gw S j x. 4. UNIFORMLY SPARSE GRAPH PROPERTIES We now introduce one specific type of matroidal graph property. DEFINITION. A graph G s Ž V, E . is uniformly k-sparse iff for every nonempty F : E, < F < y < V w F x< F k. PROPOSITION 4. The property ‘‘uniformly k-sparse’’ is matroidal for all integer k. Proof. For an arbitrary graph G s Ž V, E . and a fixed integer k, define an integer-valued function f on 2 E such that fŽ X. s
½
0, V w X x q k,
if X s ⭋, otherwise.
It can be verified then that f is nondecreasing Ži.e., A : B « f Ž A. F f Ž B .. and submodular. This in turn implies that f defines a matroid on E, where X : E is an independent set iff f Ž Y . G < Y < ᭙Y : X Žsee w17, Corollary 8.1.1x.. Let us denote by Sk Ž G . the matroid thus derived on G from the uniformly k-sparse property. For all k G y2, ‘‘uniformly k-sparse’’ is a distinct graph property, and this way an infinite sequence of hereditary properties is obtained. In fact we have already seen some examples of uniformly k-sparseness; for k s y2 Ž k s y1, resp.. it corresponds to the property ‘‘independent set’’ Ž‘‘acyclic,’’ resp... Or, in terms of matroids,
APPROXIMATING NODE-DELETION PROBLEMS
221
Sy2 Ž G . is the matroid with an empty set as its unique independent set, Sy1Ž G . is the cycle matroid, and Sk Ž G .’s for k G 0 are the so-called elongation of a cycle matroid Žsee w17, p. 60.x.. It is also easy to see that these properties satisfy the prerequisite of having infinitely many minimal forbidden graphs. PROPOSITION 5. There exist an infinite number of minimal forbidden graphs for the uniformly k-sparse property for all k G y1. Proof. Let H be a graph with two vertices and k q 3 ŽG 2. Žmultiple. edges between them. It suffices to observe now that any graph G homeomorphic to H is a minimal forbidden graph since < EŽ G .< y < V Ž G .< s k q 1 yet < EŽ G⬘.< y < V Ž G⬘.< F k for any proper subgraph G⬘ of G. The next is a key lemma in the current analysis: LEMMA 6. For any uniformly k-space property let Sk s Ž EŽ G ., r . be the matroid deri¨ ed on G from . If X is a minimal solution of NDŽ . in G
Ý
r d Ž ␦ Ž u. . F 2 ⭈ r d Ž EŽ G. . .
Ž 3.
ugX
The proof of this lemma will be given in Section 5. Finally, it is observed that, given G s Ž V, E ., we can compute efficiently Žin time O Ž< F < q < V w F x<.. the rank r Ž F . of any F : E Žand, thus, r d Ž ␦ Ž u.. for each u g V . under Sk Ž G . Žfor instance, using formula Ž5. of Lemma 10 given in Section 5.. Therefore, our algorithm PD runs in polynomial time for every Sk . Now from Lemma 6 and Theorem 3 it easily follows that THEOREM 7. When is ‘‘uniformly k-sparse’’ for any fixed k the algorithm PD computes a solution of NDŽ . in polynomial time; its performance ratio is bounded abo¨ e by 2. And hence, there exist at least countable many nontrivial hereditary properties with an infinite number of minimal forbidden graphs for which the node-deletion problems are efficiently approximable to within a factor of 2. We also deduce from Lemma 6, Ž2., and the fact that F l S is a minimal solution in Gw S x whenever yS is nonzero that the integrality gap U in our formulation is at most 2 when s‘‘uniformly k-sparse.’’ Let ZIP and U ZD be the optimal values of ŽIP2 . and ŽD 2 ., respectively, for such . And then, for any F any y computed by PD, U ZIP F
Ý ugV
F
Ý S:V
wu x uF s
Ý ugF
wu s
Ý S:V
2 r d Ž E w S x . yS F 2 ZDU .
ž
Ý ugSlF
r d Ž ␦S Ž u . . yS
/
222
TOSHIHIRO FUJITO
COROLLARY 8. When Ž IP2 . is an IP formulation of NDŽ . for uniformly U k-sparse property , ZIP rZDU F 2. 5. PROOF OF LEMMA 6 Let X be a minimal solution of NDŽ . in G s Ž V, E . for s‘‘uniformly k-sparse.’’ As preliminaries consider first the simplest case of k s y2 Ži.e., the VC problem.. Then, since r d Ž F . s < F < for any F : E, Ý u g X r d Ž ␦ Ž u .. s Ý u g X < ␦ Ž u .< s 2 < E w X x< q < E w X, V y X x< F 2 < E < s 2 r d Ž E . Žit does not matter here whether X is minimal or not.. Another particular case treated here is when k s y1 Ži.e., the FVS problem.. Assume for the simplicity that G is biconnected. Since Gw V y X x is a forest, < Ew V y X x< s < V y X < y c, where c denotes the number of components in Gw V y X x. Also, as is well known, r d Ž E . s < E < y r Ž E . s < E < y < V < q 1, and hence, r d Ž E . s < Ew X x< q < Ew X, V y X x< q Ž< V y X < y c . y < V < q 1 s < Ew X x< q < Ew X, V y X x< y < X < y c q 1. Using the characterization of the dual rank in a general matroid M we may write r d Ž ␦ Ž u . . s max ␦ Ž u . y B : B is a base of M 4 .
Ž 4.
Since B is a spanning tree of G in the present case, easily r d Ž ␦ Ž u.. F < ␦ Ž u.< y 1 for any u g V, and hence, Ý ug X r d Ž ␦ Ž u.. F 2 < Ew X x< q < Ew X, V y X x< y < X <. Now we have 2 r dŽ E. y
Ý
r d Ž ␦ Ž u . . G 2 Ž E w X x q E w X , V y X x y < X < y c q 1.
ugX
y Ž2 E w X x q E w X , V y X x y < X <. s Ew X , V y X x y Ž < X < q 2 c. q 2 and it suffices to show that < Ew X, V y X x< G < X < q 2 c. Recall that X is a minimal solution, which implies that any u g X is connected to one component of Gw V y X x in G ¨ ia two edges, so that u and V y X together induce a cycle. Since G is biconnected each component of Gw V y X x is connected to two different vertices of X in G, and it remains so even if one of such a pair of edges both connecting u and a component of Gw V y X x is removed for each u g X. Therefore, < Ew X, V y X x< G < X < q 2 c. Now in general the inequality Ž3. is trivial if < X < F 2 since r d Ž ␦ Ž u.. F dŽ . r E for any u g V in any matroid. Another trivial case occurs when E is independent, meaning that G satisfies , for then a minimal solution X must be empty and the inequality in question becomes vacuous.
APPROXIMATING NODE-DELETION PROBLEMS
223
Let G1 , . . . , Gt denote all the connected components of G. It is easy to verify that S k Ž G . is the direct sum of Sk Ž Gi .’s for any k; thus, Skd Ž G . s [1 F iF t Skd Ž Gi ., and r d Ž F . s Ý1 F iF t rid Ž F l EŽ Gi .. for any F : E, where Skd Ž Gi . s Ž EŽ Gi ., rid . is the dual of Sk Ž Gi .. Moreover, X l V Ž Gi . is a minimal solution for any Gi , and for these reasons each component can be analyzed independently. In a summary it remains to analyze the case when k G 0, assuming that < X < G 3, G is connected, and E is dependent in Sk Ž G .. 5.1. Characterizations of Sk Ž G . To study the matroid Sk Ž G . itself we introduce the concept of ‘‘surplus.’’ DEFINITIONS. For any graph G s Ž V, E . and F : E, the surplus of the subgraph Gw F x is defined by spŽ F . s < F < y < V w F x<. When C is a connected component of G let spŽ C . denote spŽ EŽ C .. s < EŽ C .< y < V Ž C .<. Let CqŽ F . and CyŽ F . denote the set of components, induced by an edge set F, with a positive surplus and the set of those with a negative surplus, respectively. When U is a vertex set of G let CqŽ U . and CyŽ U . denote CqŽ EwU x. and CyŽ EwU x., respectively. Notice that CyŽ F . is the set of all the acyclic components induced by F, each of them with a surplus of y1. Clearly, spŽ F . s ÝC g C q Ž F . spŽ C . y < CyŽ F .<. Also, LEMMA 9. For any graph G and F : EŽ G ., spŽ F . F ÝC g C q Ž F . spŽ C . F ÝC g C q ŽG. spŽ C .. When G is connected and F / ⭋, spŽ F . F spŽ G .. Proof. Certainly, spŽ F . F ÝC g C q Ž F . spŽ C .. For any component CF g C F . there is a component CG of G such that EŽ CF . : EŽ CG .. Let EF : EŽ CG . denote the union of the edge sets of those components in CqŽ F . contained entirely in CG . It then suffices to show spŽ EF . F spŽ CG .. To see it, notice that, to any edge set, adding an edge disjointly results in a decreased surplus, but ‘‘attaching’’ an edge never decreases its surplus. And of course, EF can be enlarged to EŽ CG . by always attaching edges. qŽ
Recall from the proof of Proposition 4 that any set F : E is independent in Sk Ž G . iff spŽ F⬘. F k ᭙F⬘ : F. Having Lemma 9 by now characterizing the surplus function, it is rather clear that independence of F can be determined by only those components of Gw F x with a positive surplus. More formally,
224
TOSHIHIRO FUJITO
LEMMA 10. 1. The rank function r of Sk Ž G . defined on G s Ž V, E . is gi¨ en by r Ž F . s V w F x y Cy Ž F . q min k,
½
sp Ž C .
Ý
Cg CqŽ F .
5
Ž 5.
for any F : E. 2. A subset F is an independent set of Sk Ž G . iff ÝC g C q Ž F . spŽ C . F k. Proof. Recall that Sk Ž G . was introduced by the nondecreasing and submodular function f of nonnegative integer values such that f Ž F . s < V w F x< q k if F / ⭋ Žand s 0 otherwise.; its rank function r is then given by
Ž f Ž Y . q < F y Y < . 5 Ž see w 17, Corollary 8.1.2x . ½ s min ž < F < , inf Ž V w Y x q k q < F < y < Y < . / s < F < q min ½ 0, k q inf Ž V w Y x y < Y < . 5
r Ž F . s min < F < , inf
Y:F
Y:F
Y:F
s < F < q min 0, k y sup Ž sp Ž Y . . .
½
5
Y:F
Applying Lemma 9 to Gw F x and Y : F we see that the subgraph of Gw F x yielding the maximum surplus is formed by all the components of Gw F x with a positive surplus; i.e., sup Y : F Ž spŽ F .. s ÝC g Cq Ž F . spŽ C .. Moreover, we may write in general that < F < s V w F x q sp Ž F . s V w F x y Cy Ž F . q
Ý
sp Ž C .
Cg CqŽ F .
for any F : E, and hence, r Ž F . s < F < q min 0, k y
ž
Ý
sp Ž C .
Cg CqŽ F .
s V w F x y Cy Ž F . q min k,
½
/ Ý
Cg CqŽ F .
5
sp Ž C . .
Thus, r Ž F . s < F < Ži.e., F is independent. iff ÝC g C q Ž F . spŽ C . F k. 5.2. Proof of Inequality Ž3. Below we will pay special attention to those components of Gw V y X x which are acyclic Ži.e., those in CyŽ V y X .. and let c denote the number
APPROXIMATING NODE-DELETION PROBLEMS
225
of them Ži.e., c s < CyŽ V y X .<.. Also, call such a component of Gw V y X x that is adjacent to a single vertex of X in G as a leaf components, and let c L denote the number of acyclic leaf components in Gw V y X x Žthus, 0 F c L F c .. Since G is connected and spŽ E . ) k Ž E is dependent., r Ž E . s < V < y < CyŽ E .< q k s < V < q k. Also, since Ew V y X x is an independent set def of Sk Ž G ., < Ew V y X x< s r Ž Ew V y X x. s < V y X < y c q l, where l s ÝC g C q ŽVyX . spŽ C ., and 0 - l F k by Lemma 10. Substituting these for r Ž E . and < Ew V y X x<, respectively, we have r d Ž E. s Ew X x q Ew X , V y X x q Ew V y X x y r Ž E. s Ew X x q Ew X , V y X x q Ž
Ž 6.
Recall Ž4. and, to estimate the value of r d Ž ␦ Ž u.. for u g X, we inspect how many edges among those incident to u must belong to every base B of Sk Ž G .. Take any acyclic leaf component T of Gw V y X x which is adjacent only to u Žif it exists.. Then, any base B must contain Žat least. one edge connecting u and T because B cannot induce an acyclic component in connected G, yet it must contain some edge of T. Besides those edges, one each for every acyclic leaf adjacent to u, B must contain at least one more edge from ␦ Ž u., if it exists. Otherwise, the component of Gw B x containing u is a tree at best. It does exist, however, because G is connected and cannot be a tree. Therefore, B must have at least Ž噛 of acyclic leaf components adjacent to u. q 1 many edges of ␦ Ž u. in it, and hence, r d Ž ␦ Ž u . . F ␦ Ž u . y Ž 1qŽ 噛 of acyclic leaf components adjacent to u . . . Summing up over all the vertices of X,
Ý
r d Ž ␦ Ž u. . F 2 E w X x q E w X , V y X x y < X < y cL .
Ž 7.
ugX
For every acyclic component T of Gw V y X x adjacent to u g X, pick one edge e connecting u and T, and color e red so that u, together with all of its adjacent acyclic components, forms a single tree via these red edges. X is a minimal solution and this implies that each u of X, together with V y X, induces a minimal forbidden graph, i.e., a circuit of Sk Ž G .. Let D : E be such a circuit existent in Gw u4 j Ž V y X .x. We examine how many edges of D necessarily exist in Ew u4 , V y X x besides those colored red. Since D is a circuit of S k Ž G ., spŽ D . s k q 1 by definition. Also recall l s ÝC g Cq ŽVyX . spŽ C ., which implies that every subgraph of Gw V y X x has a surplus of at most l Žby Lemma 9.. Moreover, the addition of u and all
226
TOSHIHIRO FUJITO
red edges to such a subgraph results in a subgraph with surplus of still at most l. Further addition of any single edge of no color between u and V y X can increase its surplus by at most one. Therefore, D must contain at least k q 1 y l edges from Ew u4 , V y X x other than those red edges, and this means the existence of that many edges in Ew u4 , V y X x. In other words, G k y l q 1 q Ž 噛 of acyclic components of G w V y X x ,
E u4 , V y X
adjacent to u . for each u in X, and summing up over all vertices of X, E w X , V y X x G Ž k y l q 1. < X < q
Ý Ž 噛 of acyclic components of G w V yX x , ugX
adjacent to u . G Ž k y l q 1. < X < q 2 Ž c y cL . q cL ,
Ž 8.
since every nonleaf of Gw V y X x must be adjacent to Žat least. two distinct vertices of X in G. Finally, multiplying Ž6. by 2 and subtracting Ž7. from it, 2 r dŽ E. y
Ý
r d Ž ␦ Ž u. .
ugX
G 2Ž Ew X x q Ew X , V y X x y Ž < X < q k y l q c . . y Ž2 E w X x q E w X , V y X x y < X < y cL . s E w X , V y X x y Ž < X < q 2Ž k y l . q 2 c y cL . G Ž k y l q 1. < X < y Ž < X < q 2 Ž k y l . .
Ž by Ž 8. .
s Ž k y l . Ž < X < y 2. G0
Ž for l F k and < X < G 2 . . ACKNOWLEDGMENTS
The author is very grateful to the anonymous referees for their valuable comments and suggestions.
REFERENCES 1. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy, Proof verification and hardness of approximation problems, in ‘‘Proc. 33rd Annual Symposium on Foundations of Computer Science ŽFOCS., 1992,’’ pp. 14᎐23.
APPROXIMATING NODE-DELETION PROBLEMS
227
2. V. Bafna, P. Berman, and T. Fujito, Constant ratio approximations of the weighted feedback vertex set problem for undirected graphs, in ‘‘Proc. 6th International Symposium on Algorithms and Computation, 1995,’’ pp. 142᎐151. 3. R. Bar-Yehuda and S. Even, A local-ratio theorem for approximating the weighted vertex cover problem, in ‘‘Annals of Discrete Mathematics, Vol. 25,’’ pp. 27᎐46, North-Holland, Amsterdam, 1985. 4. R. Bar-Yehuda, D. Geiger, J. Naor, and R. M. Roth, Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and Bayesian inference, in ‘‘Proc. 5th Annual ACM-SIAM Symposium on Discrete Algorithms, 1994,’’ pp. 344᎐354. 5. A. Becker and D. Geiger, Approximation algorithms for the loop cutset problem, in ‘‘Proc. 10th Conf. on Uncertainty in AI, 1994,’’ pp. 60᎐68. 6. F. A. Chudak, M. X. Goemans, D. S. Hochbaum, and D. P. Williamson, A primal-dual interpretation of recent 2-approximation algorithms for the feedback vertex set problem in undirected graphs, Oper. Res. Lett., 22 Ž1998., 111᎐118. 7. V. Chvatal, ´ A greedy heuristic for the set-covering problem, Math. Oper. Res. 4Ž3. Ž1979., 233᎐235. 8. G. Even, J. Naor, B. Schieber, and M. Sudan, Approximating minimum feedback sets and multi-cuts in directed graphs, Algorithmica 20 Ž1998., 151᎐174. 9. G. Even, J. Naor, B. Schieber, and L. Zosin, Approximating minimum subset feedback sets in undirected graphs with applications, in ‘‘4th Israel Symposium on Theory of Computing and Systems, 1996,’’ pp. 78᎐88. 10. M. R. Garey and D. S. Johnson, ‘‘Computers and Intractability: A Guide to Theory of NP-Completeness,’’ Freeman, New York, 1979. 11. N. Garg, V. Vazirani, and M. Yannakakis, Multiway cuts in directed and node weighted graphs, in ‘‘Proc. 21st International Colloquium on Automata, Languages and Programming, 1994,’’ pp. 487᎐498. 12. J. Hastad, Some optimal in-approximability results, in ‘‘Proc. 29th Annual ACM Sympo˚ sium on Theory of Computing ŽSTOC., 1997,’’ pp. 1᎐10. 13. J. M. Lewis and M. Yannakakis, The Node-deletion problem for hereditary properties is NP-complete, J. Comput. System Sci. 20 Ž1980., 219᎐230. 14. C. Lund and M. Yannakakis, The approximation of maximum subgraph problems, in ‘‘Proc. 20th International Colloquium on Automata, Languages and Programming, 1993,’’ pp. 40᎐51. 15. B. Monien and E. Speckenmeyer, Ramsey numbers and an approximation algorithm for the vertex cover problem, Acta Inform. 22 Ž1985., 115᎐123. 16. C. Papadimitriou and M. Yannakakis, Optimization, approximation and complexity classes, J. Comput. System Sci. 43 Ž1991., 425᎐440. 17. D. J. A. Welsh, ‘‘Matroid Theory,’’ Academic Press, London, 1976. 18. L. A. Wolsey, An analysis of the greedy algorithm for the submodular set covering problem, Combinatorica 2Ž4. Ž1982., 385᎐393. 19. M. Yannakakis, Some open problems in approximation, in ‘‘Proc. 2nd Italian Conference on Algorithms and Complexity, 1994,’’ pp. 33᎐39.
Approximating Node-Deletion Problems for Matroidal Properties* Toshihiro Fujito Department of Electrical Engineering, Hiroshima Uni¨ ersity, 1-4-1 Kagamiyama, Higashi-Hiroshima 739 Japan E-mail: [email protected] Received October 22, 1997; revised November 2, 1998
The paper is concerned with the polynomial time approximability of node-deletion problems for hereditary properties. It is observed that, when such a property derives a matroid on any graph, the problem can be formulated as a matroid set cover optimization problem, and this leads us naturally to consider two well-known approaches, greedy and primal-dual. The heuristics based on these principles are analyzed and general approximation bounds are obtained. Next, more specific types of hereditary properties, called uniformly sparse, are introduced and, for any of them, the primal-dual heuristic is shown to approximate the corresponding nodedeletion problem within a factor of 2. We conclude that there exist infinitely many Žat least countably many. node-deletion problems, each of which approximable to a factor of 2, for hereditary properties with an infinite number of minimal forbidden graphs. 䊚 1999 Academic Press
1. INTRODUCTION The paper is concerned with the polynomial time approximability of node-deletion problems for hereditary properties. The node-deletion problem for a graph property Ždenoted NDŽ . throughout the paper. is a typical graph optimization problem; that is, give a node-weighted graph G, find a vertex set of the minimum weight sum whose deletion Žalong with all the incident edges. from G leaves a subgraph satisfying the property . A graph property is hereditary if every subgraph of a graph satisfying * A preliminary version presented at the 24th International Colloquium on Automata, Languages and Programming, Bologna, Italy, July 1997. 211 0196-6774r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
212
TOSHIHIRO FUJITO
also satisfies . A number of well-studied graph properties are hereditary such as independent set, planar, bipartite, acyclic, degree-constrained, circular-arc, circle graph, chordal, comparability, permutation, perfect. Naturally, many well-known graph problems fall into this class of problems when desired graph properties are specified appropriately. Lewis and Yannakakis proved, however, that whenever is nontrivial and hereditary on induced subgraphs NDŽ . is NP-hard w13x. When this general result of NP-hardness was established, almost nothing was known about the polynomial time approximability of NDŽ .’s except for good approximation algorithms for the vertex cover ŽVC. problem. Moreover, their generic reductions from VC to other NDŽ .’s are approximation preserving, and as such, no NDŽ . can be approximated better than VC can be. Thus it was natural for them to pose a question: can other node-deletion problems be approximated well? It has been long known that VC can be approximated with ratio 2 Žachievable by a simple maximal matching based heuristic w10, p. 134x for unweighted graphs. and a better approximation had been a subject of extensive research over the years. Yet the best constant bound has remained the same at 2 while the best known heuristics can accomplish only slightly better Ž2 y log log nr2 log n of w3, 15x.. On the other hand, very few other NDŽ .’s are known to be approximable within some constant factor even today, not to mention a factor of 2. As observed in w14x whenever hereditary has only a finite number of minimal forbidden graphs NDŽ . can be efficiently approximated within some constant factor of the optimum. It was in fact conjectured therein that these are the only node-deletion problems approximable within some constant factor Žsee also w19x.. It was found later, however, that this conjecture does not hold, as the Žunweighted. feedback vertex set ŽFVS. problem Ži.e., s‘‘acyclic’’. on undirected graphs was shown to be approximable within a factor of 4 w4x Žnote, a simple cycle of every length is a minimal forbidden graph for this .. While even the weighted version was later found to be constantly approximable Žsee below., until now this problem has been the only known exception to the conjecture of Lund and Yannakakis. 1.1. Our Results We will show that there exist infinitely many Žat least countably many. constant factor approximable NDŽ .’s for with an infinite number of minimal forbidden graphs. For that purpose we shall concentrate on such hereditary properties that derive matroids on Žthe edge set of. any graph Ždetails given later.. The class of NDŽ .’s for such properties includes VC, FVS, and many others.
APPROXIMATING NODE-DELETION PROBLEMS
213
The problems for such ’s will be cast in a form of matroid set cover optimization problems. This reformulation leads us naturally to consider two well-known approaches, greedy and primal-dual. The heuristic based on each principle is analyzed and general approximation bounds are obtained for NDŽ .’s. It turns out that our primal-dual algorithm is simpler than those primal-dual based 2-approximation algorithms designed particularly for FVS Žsee below.. Next, to capture a class of constant factor approximable NDŽ .’s, more specific types of hereditary properties are introduced. It will be shown that, for such properties called ‘‘uniformly sparse,’’ a certain combinatorial bound holds, in terms of matroid ranks, for any minimal solution of NDŽ .. This allows us to apply the general bound of primal-dual for this class of NDŽ .’s and conclude that, for any of Žinfinitely many. uniformly sparse properties Žeach having an infinite number of minimal forbidden graphs., NDŽ . is approximable within a factor of 2, matching the best constant factor known for either VC or FVS. 1.2. Other Related Work Every NDŽ . for nontrivial hereditary is MAX SNP-hard, as pointed out in w14x, due to the reductions of w13x and the result of w16x. Thus, there exists some positive constant ⑀ such that no polynomial time algorithm for NDŽ . can achieve an approximation factor better than 1 q ⑀ , unless P s NP w1x. Yet a better lower bound is provided by the one in approximation of VC as it serves as a lower bound for every NDŽ . for hereditary . The lower bound for VC approximation has been continuously improved in the last few years and currently it is known to be arbitrarily close to 7 Ž . w12x. 6 modulo P / NP The approximation ratio of 4 for the unweighted FVS w4x was subsequently extended to the one for the weighted FVS and was further improved to 2 w2, 5x. Recently Chudak et al. gave a primal-dual interpretation of these 2-approximation algorithms w6x and also presented a new primal-dual algorithm for FVS which has the same performance ratio but is slightly simpler than the previous two. All of these 2-approximation algorithms as well as ours, based on the same principle, are similar in their execution, repeatedly reducing a certain form of weight distributions defined on a certain set of subgraphs. Our primal-dual algorithm is simpler than others, however, since it does not look into nor modify explicitly any special structure in graphs under consideration. Some nontrivial Žalbeit nonconstant. approximation factors known for other specific NDŽ .’s include O Žlog < V <. for s‘‘bipartite’’ w11x and O Žlog < V
214
TOSHIHIRO FUJITO
2. PRELIMINARIES 2.1. Notation and Definitions A subgraph of G s Ž V, E . induced by X : V is denoted by Gw X x. For X : V let Ew X x denote the set of edges induced by X, and conversely, let V w F x for F : E denote the set of vertices incident to some edge in F. Ew X, Y x is the set of edges with one end in X and the other in Y. For any graph G let V Ž G . and EŽ G . denote the vertex set and the edge set of G, respectively. The set of edges incident to some vertex of X is denoted ␦ Ž X . and when those edges are restricted to the ones in a subgraph Gw Y x we denote it by ␦ Y Ž X .Žs ␦ Ž X . l Ew Y x.. Let ␦ Ž u. Ž ␦ Y Ž u., respectively. be a shorthand of ␦ Ž u4. Ž ␦ Y Ž u4., respectively.. A graph property is nontri¨ ial if infinitely many graphs satisfy and infinitely many graphs fail to satisfy it. It is hereditary Ž on induced subgraphs. if, in any graph satisfying , every Žresp., vertex-induced. subgraph also satisfies . For a hereditary property any graph which does not satisfy is called a forbidden graph for , and it is a minimal one if, additionally, every ‘‘proper’’ Žresp., induced. subgraph of it satisfies . Any hereditary property is equivalently characterized by the set of all the minimal forbidden graphs for . It is customary to measure the quality of an approximation algorithm by its performance ratio, which is the worst case ratio of the optimal value to the value of an approximate solution returned by the algorithm for the same instance. 2.2. Matroidal Properties One way to represent a matroid M is by a pair Ž E, r ., where E is the ground set and r is the rank function defined on 2 E, of M. A set F : E is called ᎏindependent if r Ž F . s < F < Žand conversely, r Ž F⬘. is the cardinality of a largest independent subset of F⬘ for an arbitrary F⬘ : E ., ᎏdependent if r Ž F . - < F <, ᎏa base if it is a maximal Žand, hence, maximum in any matroid. independent set, ᎏa circuit if it is a minimal dependent set, ᎏspanning if r Ž F . s r Ž E .. For any matroid M s Ž E, r . there is the dual matroid M d s Ž E, r d . defined on the same ground set E. The rank functions r and r d are related such that 䢇
䢇
r dŽ E y F . s Ž < E< y r Ž E. . y Ž < F < y r Ž F . . for any F : E.
APPROXIMATING NODE-DELETION PROBLEMS
215
For two matroids M1 s Ž E1 , r 1 . and M2 s Ž E2 , r 2 . defined on disjoint sets E1 and E2 , the direct sum of M1 and M2 , written M1 [ M2 , is the matroid on E1 j E2 such that its rank function r is given by r Ž F . s r 1Ž F l E1 . q r 2 Ž F l E2 . for any subset F : E1 j E2 . It is easy to verify that the dual of the direct sum of M1 and M2 is the direct sum of the duals of M1 and M2 ; that is, Ž M1 [ M2 . d s M1d [ M2d. If I Ž M . is the set of independent sets of matroid M on E, I: I g I Ž M ., I : F 4 for any F : E is also the set of independent sets of a matroid defined on F. Such a matroid is called a restriction of M to F; the rank function here is just the rank function of M but is restricted to subsets of F. ŽFor more on matroid theory see, e.g., w17x.. 䢇
䢇
We say that a hereditary graph property is matroidal if, on any graph G s Ž V, E ., the edge sets of subgraphs of G satisfying form the set of independent sets of a matroid defined on E. Such a matroid is said to be deri¨ ed from the property on E. The property s‘‘acyclic’’ is a typical example of such properties, where the family of edge sets of acyclic subgraphs defines a cycle matroid on any graph. 䢇
Denote by I Ž G . the family of subgraphs of G satisfying and consider any subgraph H of G. Then, for any , I Ž H . is I Ž G . restricted to H, i.e., I Ž G . l G⬘: G⬘ is a subgraph of H 4 . Therefore, if is matroidal, the matroid derived from on EŽ H . is necessarily a restriction of the one derived from on EŽ G .. This means that the rank functions of matroids derived from any are identical, differing only in their domains. For this reason the rank function of a derived matroid on any Žedge set of. graph will be denoted by r Žalong with specification of its domain.. Note also that, by definition, when a matroid M is derived from on EŽ G ., a subgraph H of G is a minimal forbidden graph for iff EŽ H . is a circuit of M. 䢇
A set function f : 2 S ª ⺢ is submodular if for any subsets A, B
of S: f Ž A. q f Ž B . G f Ž A l B . q f Ž A j B . .
3. APPROXIMATION ALGORITHMS FOR MATROIDAL PROPERTIES 3.1. Formulations Perhaps a most natural integer program formulation of NDŽ ., when is general hereditary, is the following ‘‘hitting set’’ formulation: using a 0᎐1 vector x as an incidence vector to represent a feasible vertex set
216
TOSHIHIRO FUJITO
F : V, it looks like Min
Ý
wu x u
ugV
subject to:
Ý
x u G 1,
H g ⍀G Ž . ,
Ž IP1 .
ugV Ž H .
x u g 0, 1 4 ,
u g V,
where ⍀ G Ž . is the set of minimal forbidden graphs for contained as subgraphs in G. It was tried by Bar-Yehuda et al. to approximate the weighted FVS using a primal-dual strategy based on this IP for FVS and its LP dual, which resulted in a factor O Žlog < V <. approximation w4x. Moreover, it was indicated in w9x that in case of the FVS problem the linear relaxation of the formulation above introduces the integrality gap Ži.e, the ratio between the integer and fractional optima. of size as large as ⍀ Žlog < V <.. While other IP formulations were found to bound the integrality gap by a factor of 2 for the FVS problem Žsee w6x., we shall show later that there exists another formulation with integrality gap of at most 2 for many NDŽ .’s, including FVS Žsee Corollary 8.. Let us assume henceforth that is matroidal. Taking advantage of the special structure of solution sets for such NDŽ .’s, we reformulate it in the form of a matroid optimization problem, where, instead of vertices, edge sets of form ␦ Ž u. are used to represent basic constituents of an instance: Matroid formulation. Given a weighted set system Ž E, EV s ␦ Ž u.: u g V 4 , w: EV ª ⺡q. and the matroid M s Ž E, r . derived from , find EF s a subcollection EF : EV of the minimum weight sum such that jE Dug F ␦ Ž u. s ␦ Ž F . is spanning in M d. Or, more concisely, min F:V
½ Ý w : r Ž ␦ Ž F . . s r Ž E. 5 . d
d
u
ugF
The equivalence follows easily from the fact that F : V is a solution of NDŽ . iff E y ␦ Ž F . is independent in M s Ž E, r . Žwhich is equivalent to say ␦ Ž F . is spanning in M d .. Now, based on this interpretation of NDŽ ., we come up with a new IP formulation, reflecting the matroid formulation. The question here is how to replace the constraint r d Ž ␦ Ž F .. s r d ŽDug F ␦ Ž u.. s r d Ž E . by linear inequalities in x. Our solution given below is to have a linear inequality of form Ý ug S l F r d Ž ␦S Ž u.. G r d Ž Ew S x. which is valid at Gw S x for every induced subgraph Gw S x.
APPROXIMATING NODE-DELETION PROBLEMS
217
We claim that NDŽ . on graph G s Ž V, E . can be formulated by the following IP when M s Ž E, r . is the matroid derived on G from : Min
Ý
wu x u
ugV
subject to:
Ý r d Ž ␦S Ž u . . x u G r d Ž E w S x . ,
S : V,
Ž IP2 .
ugS
x u g 0, 1 4 ,
u g V.
THEOREM 1. When M s Ž E, r . is the matroid deri¨ ed on G from , F is a solution of NDŽ . iff x F g 0, 14V Ž incidence ¨ ector of F . is a feasible solution to Ž IP2 .. Proof. ŽOnly if. F : V is a solution in G m F l S is a solution in G w S x
᭙S : V
m ␦S Ž FlS . is spanning in M d Ž G w S x . m r d Ž ␦S Ž F l S . . s r d Ž E w S x . d
Because r is a nonnegative submodular function Ý i r any family of Ei ’s, and hence, ᭙S : V,
Ý r d Ž ␦S Ž u . . x uF s Ý ugS
dŽ
᭙S:V
᭙S : V .
Ei . G r d ŽDi Ei . for
r d Ž ␦S Ž u . . G r d Ž ␦S Ž F l S . . s r d Ž E w S x . .
ugFlS
ŽIf. Suppose F is not a solution of NDŽ .. This implies the existence of a minimal forbidden graph H for , as a subgraph of G, avoided by F. Since F l V Ž H . s ⭋,
Ý
r d Ž ␦ V Ž H . Ž u . . x uF s 0.
ugV Ž H .
Since EŽ H . is a circuit of M Ž G . Žand of M Ž H .., V Ž H . induces an edge set which is dependent in M Ž G . Žand in M Ž H ... This implies r Ž Ew V Ž H .x. < Ew V Ž H .x<, and hence, rdŽ E VŽ H .
.s
E VŽ H .
y rŽ E VŽ H .
. ) 0.
Thus, x F does not satisfy the constraint corresponding to V Ž H .. 3.2. Greedy Heuristic Given a Žmatroid. set system based formulation and its resemblance to the standard set cover problem, the greedy heuristic is an approach to come in due course. In the current context such a heuristic, called
218
TOSHIHIRO FUJITO
FIG. 1. Greedy algorithm Greedy for NDŽ ..
Greedy, repeatedly selects a subset Žof form ␦ Ž u.. for inclusion, which gives a maximal increase in the rank of a current solution per weight Žsee Fig. 1.. Wolsey showed that the well-known analysis of Chvatal ´ w7x for greedy set cover approximation generalizes to the case of a matroid set system w18x Žactually, even to more general submodular systems.; that is, PROPOSITION 2. Let H Ž i . denote the ith Harmonic number, and ⌬ s max ug V r d Ž ␦ Ž u... On any graph G s Ž V, E . Greedy finds a solution of NDŽ . with its weight bounded by H Ž ⌬ . F 1 q ln ⌬ times the optimum. def
Thus, Greedy approximates NDŽ . within an O Žlog n. factor for any matroidal Žinterestingly, as mentioned earlier, the ŽIP1 .-based primaldual heuristic for FVS achieves a tight performance ratio of the same order.. One serious drawback of Greedy, however, is that it cannot deliver any constant factor approximation in general, even for the simplest case of VC problem. 3.3. Primal-Dual Heuristic We now turn to an alternative approach, primal-dual, and we do so using the IP formulation ŽIP2 .. Take first the dual of the LP relaxation of ŽIP2 . Žin which every integrality constraint on x u is replaced by x u G 0.: Max
Ý r d Ž E w S x . yS S:V
subject to:
Ý
r d Ž ␦S Ž u . . yS F wu ,
u g V,
Ž D2 .
S: ugS
yS G 0,
S : V.
The primal-dual approximation algorithm PD, based on ŽIP2 . and ŽD 2 . is presented in Fig. 2. We elaborate more on it. PD starts with F s ⭋, the original graph Gw S⬘x s Ž V, E . and the dual feasible solution y s 0. Given F, if it is not yet a solution of NDŽ . there must exist some set S : V
APPROXIMATING NODE-DELETION PROBLEMS
219
FIG. 2. Primal-Dual Algorithm PD for NDŽ ..
corresponding to a violated constraint of ŽIP2 .. In particular the set of all the remaining vertices S⬘Žs V y F . must be always such a set, and thus we can always choose S⬘ as a ‘‘violated set.’’ PD then increases the dual variable yS⬘ as much as possible until for some vertex u in S⬘ the dual constraint for u becomes tight; i.e., r d Ž ␦S Ž u . . yS s wu .
Ý
Ž 1.
S: ugS
Notice that yS⬘ here can be increased indeed because S⬘ is the collection of all those vertices whose corresponding dual constraints were not yet tight. PD adds u into a solution set F and at the same time removes it from S⬘. Clearly F eventually becomes a solution of NDŽ . Žand to ŽIP2 .. while y is kept feasible to ŽD 2 .. Lastly, the vertices in F are examined one by one, in th reverse order of their inclusion to F, and whenever any of them is found to be extraneous it is thrown out of F. PD clearly constructs a feasible solution F of NDŽ . and a solution y feasible for ŽD 2 .. These two solution are related, by means of Ž1., such that
Ý ugF
wu s
r d Ž ␦S Ž u . . ys s
Ý Ý ugF S : ugS
Ý S:V
ž
Ý ugSlF
r d Ž ␦S Ž u . . yS .
/
Ž 2.
The analysis of the performance ratio of PD can be reduced to that of the following combinatorial bound. THEOREM 3. Let M s Ž EŽ G ., r . be the matroid deri¨ ed from . Then the performance ratio of the primal-dual algorithm PD is bounded by max
½
Ý ug X r d Ž ␦ Ž u . . r d Ž EŽ G. .
5
,
where max is taken o¨ er any minimal solution X of NDŽ . in any G.
220
TOSHIHIRO FUJITO
Proof. Let F denote the output solution of PD. It follows from Ž2. Žand the form of the objective function in ŽD 2 .. that the approximation ratio is reduced to the maximum ratio between Ý ug S l F r d Ž ␦S Ž u.. and r d Ž Ew S x. for any S with nonzero yS . Let Sl be S⬘ chosen by the lth iteration of the While-loop; that is, Sl s V y u1 , . . . , u ly14 , and they are the only S’s with yS ) 0. Thus, it suffices to show that Sl l F is a minimal solution in Gw Sl x for all l G 1. Recall that the last clean-up step of the algorithm examines, in the decreasing order of l, if u l ’s are needed for F to be a solution in G. Let Fj b F remaining after all u l ’s, for l G j, are examined by this For-loop. Then, S j l F s S j l Fj for all j G 1. Assume inductively that Sl l Fl is a minimal solution in Gw Sl x for all l G j q 1. Now if u j f Fj it is clear that Fl l Sl is a minimal solution in Gw Sl x for all l G j. Otherwise, u j g Fj because Fj y u j 4 is not a solution in G, but it implies that neither is Ž S j l Fj . y u j 4 in Gw S j x. Therefore, S j l Fj Žs S j l F . must be a minimal solution in Gw S j x. 4. UNIFORMLY SPARSE GRAPH PROPERTIES We now introduce one specific type of matroidal graph property. DEFINITION. A graph G s Ž V, E . is uniformly k-sparse iff for every nonempty F : E, < F < y < V w F x< F k. PROPOSITION 4. The property ‘‘uniformly k-sparse’’ is matroidal for all integer k. Proof. For an arbitrary graph G s Ž V, E . and a fixed integer k, define an integer-valued function f on 2 E such that fŽ X. s
½
0, V w X x q k,
if X s ⭋, otherwise.
It can be verified then that f is nondecreasing Ži.e., A : B « f Ž A. F f Ž B .. and submodular. This in turn implies that f defines a matroid on E, where X : E is an independent set iff f Ž Y . G < Y < ᭙Y : X Žsee w17, Corollary 8.1.1x.. Let us denote by Sk Ž G . the matroid thus derived on G from the uniformly k-sparse property. For all k G y2, ‘‘uniformly k-sparse’’ is a distinct graph property, and this way an infinite sequence of hereditary properties is obtained. In fact we have already seen some examples of uniformly k-sparseness; for k s y2 Ž k s y1, resp.. it corresponds to the property ‘‘independent set’’ Ž‘‘acyclic,’’ resp... Or, in terms of matroids,
APPROXIMATING NODE-DELETION PROBLEMS
221
Sy2 Ž G . is the matroid with an empty set as its unique independent set, Sy1Ž G . is the cycle matroid, and Sk Ž G .’s for k G 0 are the so-called elongation of a cycle matroid Žsee w17, p. 60.x.. It is also easy to see that these properties satisfy the prerequisite of having infinitely many minimal forbidden graphs. PROPOSITION 5. There exist an infinite number of minimal forbidden graphs for the uniformly k-sparse property for all k G y1. Proof. Let H be a graph with two vertices and k q 3 ŽG 2. Žmultiple. edges between them. It suffices to observe now that any graph G homeomorphic to H is a minimal forbidden graph since < EŽ G .< y < V Ž G .< s k q 1 yet < EŽ G⬘.< y < V Ž G⬘.< F k for any proper subgraph G⬘ of G. The next is a key lemma in the current analysis: LEMMA 6. For any uniformly k-space property let Sk s Ž EŽ G ., r . be the matroid deri¨ ed on G from . If X is a minimal solution of NDŽ . in G
Ý
r d Ž ␦ Ž u. . F 2 ⭈ r d Ž EŽ G. . .
Ž 3.
ugX
The proof of this lemma will be given in Section 5. Finally, it is observed that, given G s Ž V, E ., we can compute efficiently Žin time O Ž< F < q < V w F x<.. the rank r Ž F . of any F : E Žand, thus, r d Ž ␦ Ž u.. for each u g V . under Sk Ž G . Žfor instance, using formula Ž5. of Lemma 10 given in Section 5.. Therefore, our algorithm PD runs in polynomial time for every Sk . Now from Lemma 6 and Theorem 3 it easily follows that THEOREM 7. When is ‘‘uniformly k-sparse’’ for any fixed k the algorithm PD computes a solution of NDŽ . in polynomial time; its performance ratio is bounded abo¨ e by 2. And hence, there exist at least countable many nontrivial hereditary properties with an infinite number of minimal forbidden graphs for which the node-deletion problems are efficiently approximable to within a factor of 2. We also deduce from Lemma 6, Ž2., and the fact that F l S is a minimal solution in Gw S x whenever yS is nonzero that the integrality gap U in our formulation is at most 2 when s‘‘uniformly k-sparse.’’ Let ZIP and U ZD be the optimal values of ŽIP2 . and ŽD 2 ., respectively, for such . And then, for any F any y computed by PD, U ZIP F
Ý ugV
F
Ý S:V
wu x uF s
Ý ugF
wu s
Ý S:V
2 r d Ž E w S x . yS F 2 ZDU .
ž
Ý ugSlF
r d Ž ␦S Ž u . . yS
/
222
TOSHIHIRO FUJITO
COROLLARY 8. When Ž IP2 . is an IP formulation of NDŽ . for uniformly U k-sparse property , ZIP rZDU F 2. 5. PROOF OF LEMMA 6 Let X be a minimal solution of NDŽ . in G s Ž V, E . for s‘‘uniformly k-sparse.’’ As preliminaries consider first the simplest case of k s y2 Ži.e., the VC problem.. Then, since r d Ž F . s < F < for any F : E, Ý u g X r d Ž ␦ Ž u .. s Ý u g X < ␦ Ž u .< s 2 < E w X x< q < E w X, V y X x< F 2 < E < s 2 r d Ž E . Žit does not matter here whether X is minimal or not.. Another particular case treated here is when k s y1 Ži.e., the FVS problem.. Assume for the simplicity that G is biconnected. Since Gw V y X x is a forest, < Ew V y X x< s < V y X < y c, where c denotes the number of components in Gw V y X x. Also, as is well known, r d Ž E . s < E < y r Ž E . s < E < y < V < q 1, and hence, r d Ž E . s < Ew X x< q < Ew X, V y X x< q Ž< V y X < y c . y < V < q 1 s < Ew X x< q < Ew X, V y X x< y < X < y c q 1. Using the characterization of the dual rank in a general matroid M we may write r d Ž ␦ Ž u . . s max ␦ Ž u . y B : B is a base of M 4 .
Ž 4.
Since B is a spanning tree of G in the present case, easily r d Ž ␦ Ž u.. F < ␦ Ž u.< y 1 for any u g V, and hence, Ý ug X r d Ž ␦ Ž u.. F 2 < Ew X x< q < Ew X, V y X x< y < X <. Now we have 2 r dŽ E. y
Ý
r d Ž ␦ Ž u . . G 2 Ž E w X x q E w X , V y X x y < X < y c q 1.
ugX
y Ž2 E w X x q E w X , V y X x y < X <. s Ew X , V y X x y Ž < X < q 2 c. q 2 and it suffices to show that < Ew X, V y X x< G < X < q 2 c. Recall that X is a minimal solution, which implies that any u g X is connected to one component of Gw V y X x in G ¨ ia two edges, so that u and V y X together induce a cycle. Since G is biconnected each component of Gw V y X x is connected to two different vertices of X in G, and it remains so even if one of such a pair of edges both connecting u and a component of Gw V y X x is removed for each u g X. Therefore, < Ew X, V y X x< G < X < q 2 c. Now in general the inequality Ž3. is trivial if < X < F 2 since r d Ž ␦ Ž u.. F dŽ . r E for any u g V in any matroid. Another trivial case occurs when E is independent, meaning that G satisfies , for then a minimal solution X must be empty and the inequality in question becomes vacuous.
APPROXIMATING NODE-DELETION PROBLEMS
223
Let G1 , . . . , Gt denote all the connected components of G. It is easy to verify that S k Ž G . is the direct sum of Sk Ž Gi .’s for any k; thus, Skd Ž G . s [1 F iF t Skd Ž Gi ., and r d Ž F . s Ý1 F iF t rid Ž F l EŽ Gi .. for any F : E, where Skd Ž Gi . s Ž EŽ Gi ., rid . is the dual of Sk Ž Gi .. Moreover, X l V Ž Gi . is a minimal solution for any Gi , and for these reasons each component can be analyzed independently. In a summary it remains to analyze the case when k G 0, assuming that < X < G 3, G is connected, and E is dependent in Sk Ž G .. 5.1. Characterizations of Sk Ž G . To study the matroid Sk Ž G . itself we introduce the concept of ‘‘surplus.’’ DEFINITIONS. For any graph G s Ž V, E . and F : E, the surplus of the subgraph Gw F x is defined by spŽ F . s < F < y < V w F x<. When C is a connected component of G let spŽ C . denote spŽ EŽ C .. s < EŽ C .< y < V Ž C .<. Let CqŽ F . and CyŽ F . denote the set of components, induced by an edge set F, with a positive surplus and the set of those with a negative surplus, respectively. When U is a vertex set of G let CqŽ U . and CyŽ U . denote CqŽ EwU x. and CyŽ EwU x., respectively. Notice that CyŽ F . is the set of all the acyclic components induced by F, each of them with a surplus of y1. Clearly, spŽ F . s ÝC g C q Ž F . spŽ C . y < CyŽ F .<. Also, LEMMA 9. For any graph G and F : EŽ G ., spŽ F . F ÝC g C q Ž F . spŽ C . F ÝC g C q ŽG. spŽ C .. When G is connected and F / ⭋, spŽ F . F spŽ G .. Proof. Certainly, spŽ F . F ÝC g C q Ž F . spŽ C .. For any component CF g C F . there is a component CG of G such that EŽ CF . : EŽ CG .. Let EF : EŽ CG . denote the union of the edge sets of those components in CqŽ F . contained entirely in CG . It then suffices to show spŽ EF . F spŽ CG .. To see it, notice that, to any edge set, adding an edge disjointly results in a decreased surplus, but ‘‘attaching’’ an edge never decreases its surplus. And of course, EF can be enlarged to EŽ CG . by always attaching edges. qŽ
Recall from the proof of Proposition 4 that any set F : E is independent in Sk Ž G . iff spŽ F⬘. F k ᭙F⬘ : F. Having Lemma 9 by now characterizing the surplus function, it is rather clear that independence of F can be determined by only those components of Gw F x with a positive surplus. More formally,
224
TOSHIHIRO FUJITO
LEMMA 10. 1. The rank function r of Sk Ž G . defined on G s Ž V, E . is gi¨ en by r Ž F . s V w F x y Cy Ž F . q min k,
½
sp Ž C .
Ý
Cg CqŽ F .
5
Ž 5.
for any F : E. 2. A subset F is an independent set of Sk Ž G . iff ÝC g C q Ž F . spŽ C . F k. Proof. Recall that Sk Ž G . was introduced by the nondecreasing and submodular function f of nonnegative integer values such that f Ž F . s < V w F x< q k if F / ⭋ Žand s 0 otherwise.; its rank function r is then given by
Ž f Ž Y . q < F y Y < . 5 Ž see w 17, Corollary 8.1.2x . ½ s min ž < F < , inf Ž V w Y x q k q < F < y < Y < . / s < F < q min ½ 0, k q inf Ž V w Y x y < Y < . 5
r Ž F . s min < F < , inf
Y:F
Y:F
Y:F
s < F < q min 0, k y sup Ž sp Ž Y . . .
½
5
Y:F
Applying Lemma 9 to Gw F x and Y : F we see that the subgraph of Gw F x yielding the maximum surplus is formed by all the components of Gw F x with a positive surplus; i.e., sup Y : F Ž spŽ F .. s ÝC g Cq Ž F . spŽ C .. Moreover, we may write in general that < F < s V w F x q sp Ž F . s V w F x y Cy Ž F . q
Ý
sp Ž C .
Cg CqŽ F .
for any F : E, and hence, r Ž F . s < F < q min 0, k y
ž
Ý
sp Ž C .
Cg CqŽ F .
s V w F x y Cy Ž F . q min k,
½
/ Ý
Cg CqŽ F .
5
sp Ž C . .
Thus, r Ž F . s < F < Ži.e., F is independent. iff ÝC g C q Ž F . spŽ C . F k. 5.2. Proof of Inequality Ž3. Below we will pay special attention to those components of Gw V y X x which are acyclic Ži.e., those in CyŽ V y X .. and let c denote the number
APPROXIMATING NODE-DELETION PROBLEMS
225
of them Ži.e., c s < CyŽ V y X .<.. Also, call such a component of Gw V y X x that is adjacent to a single vertex of X in G as a leaf components, and let c L denote the number of acyclic leaf components in Gw V y X x Žthus, 0 F c L F c .. Since G is connected and spŽ E . ) k Ž E is dependent., r Ž E . s < V < y < CyŽ E .< q k s < V < q k. Also, since Ew V y X x is an independent set def of Sk Ž G ., < Ew V y X x< s r Ž Ew V y X x. s < V y X < y c q l, where l s ÝC g C q ŽVyX . spŽ C ., and 0 - l F k by Lemma 10. Substituting these for r Ž E . and < Ew V y X x<, respectively, we have r d Ž E. s Ew X x q Ew X , V y X x q Ew V y X x y r Ž E. s Ew X x q Ew X , V y X x q Ž
Ž 6.
Recall Ž4. and, to estimate the value of r d Ž ␦ Ž u.. for u g X, we inspect how many edges among those incident to u must belong to every base B of Sk Ž G .. Take any acyclic leaf component T of Gw V y X x which is adjacent only to u Žif it exists.. Then, any base B must contain Žat least. one edge connecting u and T because B cannot induce an acyclic component in connected G, yet it must contain some edge of T. Besides those edges, one each for every acyclic leaf adjacent to u, B must contain at least one more edge from ␦ Ž u., if it exists. Otherwise, the component of Gw B x containing u is a tree at best. It does exist, however, because G is connected and cannot be a tree. Therefore, B must have at least Ž噛 of acyclic leaf components adjacent to u. q 1 many edges of ␦ Ž u. in it, and hence, r d Ž ␦ Ž u . . F ␦ Ž u . y Ž 1qŽ 噛 of acyclic leaf components adjacent to u . . . Summing up over all the vertices of X,
Ý
r d Ž ␦ Ž u. . F 2 E w X x q E w X , V y X x y < X < y cL .
Ž 7.
ugX
For every acyclic component T of Gw V y X x adjacent to u g X, pick one edge e connecting u and T, and color e red so that u, together with all of its adjacent acyclic components, forms a single tree via these red edges. X is a minimal solution and this implies that each u of X, together with V y X, induces a minimal forbidden graph, i.e., a circuit of Sk Ž G .. Let D : E be such a circuit existent in Gw u4 j Ž V y X .x. We examine how many edges of D necessarily exist in Ew u4 , V y X x besides those colored red. Since D is a circuit of S k Ž G ., spŽ D . s k q 1 by definition. Also recall l s ÝC g Cq ŽVyX . spŽ C ., which implies that every subgraph of Gw V y X x has a surplus of at most l Žby Lemma 9.. Moreover, the addition of u and all
226
TOSHIHIRO FUJITO
red edges to such a subgraph results in a subgraph with surplus of still at most l. Further addition of any single edge of no color between u and V y X can increase its surplus by at most one. Therefore, D must contain at least k q 1 y l edges from Ew u4 , V y X x other than those red edges, and this means the existence of that many edges in Ew u4 , V y X x. In other words, G k y l q 1 q Ž 噛 of acyclic components of G w V y X x ,
E u4 , V y X
adjacent to u . for each u in X, and summing up over all vertices of X, E w X , V y X x G Ž k y l q 1. < X < q
Ý Ž 噛 of acyclic components of G w V yX x , ugX
adjacent to u . G Ž k y l q 1. < X < q 2 Ž c y cL . q cL ,
Ž 8.
since every nonleaf of Gw V y X x must be adjacent to Žat least. two distinct vertices of X in G. Finally, multiplying Ž6. by 2 and subtracting Ž7. from it, 2 r dŽ E. y
Ý
r d Ž ␦ Ž u. .
ugX
G 2Ž Ew X x q Ew X , V y X x y Ž < X < q k y l q c . . y Ž2 E w X x q E w X , V y X x y < X < y cL . s E w X , V y X x y Ž < X < q 2Ž k y l . q 2 c y cL . G Ž k y l q 1. < X < y Ž < X < q 2 Ž k y l . .
Ž by Ž 8. .
s Ž k y l . Ž < X < y 2. G0
Ž for l F k and < X < G 2 . . ACKNOWLEDGMENTS
The author is very grateful to the anonymous referees for their valuable comments and suggestions.
REFERENCES 1. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy, Proof verification and hardness of approximation problems, in ‘‘Proc. 33rd Annual Symposium on Foundations of Computer Science ŽFOCS., 1992,’’ pp. 14᎐23.
APPROXIMATING NODE-DELETION PROBLEMS
227
2. V. Bafna, P. Berman, and T. Fujito, Constant ratio approximations of the weighted feedback vertex set problem for undirected graphs, in ‘‘Proc. 6th International Symposium on Algorithms and Computation, 1995,’’ pp. 142᎐151. 3. R. Bar-Yehuda and S. Even, A local-ratio theorem for approximating the weighted vertex cover problem, in ‘‘Annals of Discrete Mathematics, Vol. 25,’’ pp. 27᎐46, North-Holland, Amsterdam, 1985. 4. R. Bar-Yehuda, D. Geiger, J. Naor, and R. M. Roth, Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and Bayesian inference, in ‘‘Proc. 5th Annual ACM-SIAM Symposium on Discrete Algorithms, 1994,’’ pp. 344᎐354. 5. A. Becker and D. Geiger, Approximation algorithms for the loop cutset problem, in ‘‘Proc. 10th Conf. on Uncertainty in AI, 1994,’’ pp. 60᎐68. 6. F. A. Chudak, M. X. Goemans, D. S. Hochbaum, and D. P. Williamson, A primal-dual interpretation of recent 2-approximation algorithms for the feedback vertex set problem in undirected graphs, Oper. Res. Lett., 22 Ž1998., 111᎐118. 7. V. Chvatal, ´ A greedy heuristic for the set-covering problem, Math. Oper. Res. 4Ž3. Ž1979., 233᎐235. 8. G. Even, J. Naor, B. Schieber, and M. Sudan, Approximating minimum feedback sets and multi-cuts in directed graphs, Algorithmica 20 Ž1998., 151᎐174. 9. G. Even, J. Naor, B. Schieber, and L. Zosin, Approximating minimum subset feedback sets in undirected graphs with applications, in ‘‘4th Israel Symposium on Theory of Computing and Systems, 1996,’’ pp. 78᎐88. 10. M. R. Garey and D. S. Johnson, ‘‘Computers and Intractability: A Guide to Theory of NP-Completeness,’’ Freeman, New York, 1979. 11. N. Garg, V. Vazirani, and M. Yannakakis, Multiway cuts in directed and node weighted graphs, in ‘‘Proc. 21st International Colloquium on Automata, Languages and Programming, 1994,’’ pp. 487᎐498. 12. J. Hastad, Some optimal in-approximability results, in ‘‘Proc. 29th Annual ACM Sympo˚ sium on Theory of Computing ŽSTOC., 1997,’’ pp. 1᎐10. 13. J. M. Lewis and M. Yannakakis, The Node-deletion problem for hereditary properties is NP-complete, J. Comput. System Sci. 20 Ž1980., 219᎐230. 14. C. Lund and M. Yannakakis, The approximation of maximum subgraph problems, in ‘‘Proc. 20th International Colloquium on Automata, Languages and Programming, 1993,’’ pp. 40᎐51. 15. B. Monien and E. Speckenmeyer, Ramsey numbers and an approximation algorithm for the vertex cover problem, Acta Inform. 22 Ž1985., 115᎐123. 16. C. Papadimitriou and M. Yannakakis, Optimization, approximation and complexity classes, J. Comput. System Sci. 43 Ž1991., 425᎐440. 17. D. J. A. Welsh, ‘‘Matroid Theory,’’ Academic Press, London, 1976. 18. L. A. Wolsey, An analysis of the greedy algorithm for the submodular set covering problem, Combinatorica 2Ž4. Ž1982., 385᎐393. 19. M. Yannakakis, Some open problems in approximation, in ‘‘Proc. 2nd Italian Conference on Algorithms and Complexity, 1994,’’ pp. 33᎐39.