Journal of Combinatorial Optimization, 5, 317 326, 2001 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. A2 1 -Approximation Algorithm for a Generalization of the Weighted Edge-Dominating Set Problem ROBERT CARR Sandia National Laboratory, P.O. Box 5800, Albuquerque, NM 87185, USA bobcarr@cs.sandia.gov TOSHIHIRO FUJITO fujito@nuee.nagoya-u.ac.jp Department of Electronics, Nagoya University Furo, Chikusa, Nagoya, 464-8603, Japan GORAN KONJEVOD konjevod@andrew.cmu.edu OJAS PAREKH odp@andrew.cmu.edu Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA, 15213-3890, USA Received August 14, 2000; Revised August 14, 2000; Accepted August 30, 2000 Abstract. We study the approximability of the weighted edge-dominating set problem. Although even the unweighted case is NP-Complete, in this case a solution of size at most twice the minimum can be efficiently computed due to its close relationship with minimum maximal matching; however, in the weighted case such a nice relationship is not known to exist. In this paper, after showing that weighted edge domination is as hard to approximate as the well studied weighted vertex cover problem, we consider a natural strategy, reducing edge-dominating set to edge cover. Our main result is a simple 2 1 -approximation algorithm for the weighted edge-dominating set problem, improving the existing ratio, due to a simple reduction to weighted vertex cover, of 2r WVC, where r WVC is the approximation guarantee of any polynomial-time weighted vertex cover algorithm. log log V The best value of r WVC currently stands at 2 2 log V. Furthermore we establish that the factor of 2 1 is tight in the sense that it coincides with the integrality gap incurred by a natural linear programming relaxation of the problem. Keywords: approximation algorithm, edge-dominating set, vertex cover, edge cover 1. Introduction In an undirected graph G = (V, E), E is a set of edges, {u,v}, where u,v belong to the set of vertices, V. An edge e dominates all f E such that e f. A set of edges is an edge-dominating set (eds) if its members collectively dominate all the edges in E. The edge-dominating set problem (EDS) is then that of finding a minimum-cardinality edgedominating set, or if edges are weighted by a function w : E Q +, an edge-dominating set of minimum total weight. Work supported in part by the United States Department of Energy under Contract DE-AC04-94AL85000. Supported in part by an NSF CAREER Grant CCR-9625297.
318 CARR ET AL. 1.1. Notation A vertex v dominates u V if {u,v} E. A vertex v also covers all edges incident upon v,or more formally, v covers an edge e if v e. We overload terminology once again and say that an edge e covers a vertex v if v e. We denote the set of edges that v covers by δ(v). When we wish to discuss the vertices of an edge set S E,wedefine V (S) = e S e.amatching is a set of edges M, such that distinct edges e, f in M do not intersect. A maximal matching is one which is not properly contained in any other matching. For S V, we denote the set {e E e S = 1} by δ(s), and we denote the set {e E e S = 2} by E(S). When given a subset S E and a vector x Q E whose components correspond to the edges in E,we use x(s), as a shorthand for e S x e. Analogously in the case of a function w v : V Q or w : E Q we write w v (S) = u S wv (u) or w(s) = e S w(e), where S V or S E, respectively. 1.2. Related problems Yannakakis and Gavril showed that EDS and the minimum maximal matching problem, whose connection to EDS will be presented later, are NP-complete even on graphs which are planar or bipartite of maximum degree 3 (Yannakakis and Gavril, 1980). This result was later extended by Horton and Kilakos to planar bipartite, line, total, perfect claw-free, and planar cubic graphs (Horton and Kilakos, 1993). On the other hand polynomially solvable special cases have been discovered. Chronologically by discovery, efficient exact algorithms for trees (Mitchell and Hedetniemi, 1977), claw-free chordal graphs, locally connected clawfree graphs, the line graphs of total graphs, the line graphs of chordal graphs (Horton and Kilakos, 1993), bipartite permutation graphs, cotriangulated graphs (Srinivasan et al., 1995), and other classes are known. Although EDS has important applications in areas such as telephone switching networks, very little is known about the weighted version of the problem. In fact, all the polynomialtime solvable cases listed above apply only to the cardinality case, although we should note that PTAS s are known for weighted planar (Baker, 1994) and λ-precision unit disk graphs (Hunt et al., 1994). In particular, while it is a simple matter to compute an edge-dominating set of size at most twice the minimum, as any maximal matching will do, such a simple reduction easily fails when arbitrary weights are assigned to edges. In fact the only known approximability result, which follows from a simple reduction to vertex cover, does not seem to have appeared in the literature. The edge-dominating set problem, especially the weighted version, seems to be the least studied among the other basic specializations of the set cover problem for graphs. The others are called the (weighted) edge cover (EC ), vertex cover (VC ), and (vertex) dominating set problems in which we seek to obtain a minimum-cardinality (weight) set which covers vertices by edges, edges by vertices, and vertices by vertices respectively. Of these only the weighted edge cover problem is known to be solvable in polynomial time (Edmonds and Johnson, 1970; Murty and Perin, 1982; Pulleyblank, 1995). Better known and studied is the dominating set problem. EDS for G is equivalent to the vertex-dominating set problem for the line graph of G. The dominating set problem for general graphs is, unfortunately,
WEIGHTED EDGE-DOMINATING SET PROBLEM 319 equivalent to the set cover problem under an approximation preserving reduction. Although the polynomial-time approximability of set cover is well established and stands at a factor of ln V +1 (Chvátal, 1979; Johnson, 1974; Lovász, 1975), it cannot be efficiently approximated better than ln V unless NP DTIME( V O(log log V ) ) (Feige, 1996). The vertex cover problem seems to be the best studied of the bunch and boasts a vast literature. Most known facts and relevant references can be found in the survey by Hochbaum (1997). The log log V 2 log V best known approximation ratio is 2, and it has been conjectured (see the above survey) that 2 is the best constant approximation factor possible in polynomial time. In this paper we consider a natural strategy of reducing weighted EDS to the related weighted edge cover problem and establish the approximability of EDS within a factor of 2 1. We also obtain the same ratio for the extension in which only a subset of the edges need be dominated. Furthermore the factor of 2 1 is tight in the sense that it coincides with the integrality gap incurred by a natural linear programming relaxation of EDS. 2. Approximation hardness Yannakakis and Gavril proved the NP-hardness of EDS by reducing VC to it (Yannakakis and Gavril, 1980). Although their reduction can be made to preserve approximation quality within some constant factor and thus imply the MAX SNP-hardness of (unweighted) EDS and the non-existence of a polynomial-time approximation scheme (unless P = NP) (Arora et al., 1992; Papadimitriou and Yannakakis, 1991), it does not preclude the possibility of better approximation of EDS than that of VC. On the other hand, it is quite straightforward to see that the approximation of weighted EDS is as hard as that of weighted VC. Theorem 1. Weighted VC can be approximated as well as weighted EDS. Proof: Let G = (V, E) be an instance graph for VC with weight function w v : V Q +. Let s be a new vertex not in V, and construct a new graph G = (V {s}, E E ) by attaching s to each vertex of G, that is, E ={{s, u} u V }. Assign a weight function w : E Q + to the edges of G by defining w (e) = w v (u) if e ={s, u} E, and w (e) = w v (u) + w v (v) if e ={u, v} E. By the definition of w, if an edge-dominating set D for G contains {u, v} E, it can be replaced by the two edges {u, s}, {v, s} E without increasing the weight of D, so we may assume D E. In this case, however, there exists a one-to-one correspondence between vertex covers in G and edge-dominating sets in G, namely C def = V (D)\{s} in G and D in G, such that w v (C) = w (D). 3. Previous work 3.1. Cardinality EDS: Reduction to maximal matching Obtaining a 2-approximation for the minimum-cardinality edge-dominating set is easy; the following proposition also demonstrates the equivalence of cardinality EDS and minimummaximal matching.
320 CARR ET AL. Proposition 1 (Harary (1969)). which is a maximal matching. There exists a minimum-cardinality edge-dominating set Proof: For a set E of edges, let adj(e ) denote the number of (unordered) pairs of adjacent edges in E, that is adj(e ) = 1 2 {(e, f ) e, f E and e f }. Let D E be a minimum-cardinality edge-dominating set. Suppose D is not a matching and let e, f D be two adjacent edges, i.e. e f. Since D is minimal, D\ f is not an edge-dominating set. Therefore there exists an edge g E adjacent to f, but not to any other member of D. Now the set D = D\{ f } {g} is another minimum-cardinality edgedominating set and adj(d )<adj(d). By repeating this exchange procedure on D,we eventually find a minimum edge-dominating set D which is a (maximal) matching. Proposition 2. Every maximal matching M gives a 2-approximation for the edge-dominating set problem. Proof: Let M 1 and M 2 be maximal matchings. The symmetric difference M 1 M 2 consists of disjoint paths and cycles in which edges alternate between those from M 1 and those from M 2. This implies an equal number of edges from M 1 and M 2 in every cycle. By the maximality of M 1 and M 2, every path must contain an edge from each of M 1 and M 2, hence every path contains at most twice as many edges from one as from the other. Letting k i = M i (M 1 M 2 ) for i = 1, 2, we now have k 1 2k 2 and k 2 2k 1. Since M i = M 1 M 2 +k i, it follows that M 1 2 M 2 and M 2 2 M 1. 3.2. Weighted EDS: Reduction to vertex cover Weighted EDS may be reformulated as finding a set of edges D of minimum weight such that V (D) is a vertex cover of G. This idea leads to a well known 2r WVC -approximation algorithm, where r WVC is the approximation guarantee of any polynomial-time weighted vertex cover algorithm. Theorem 2 (Folklore). The weighted edge-dominating set problem can be approximated to within a factor of 2r WVC. Proof: Given an instance of weighted EDS, G with weight function w : E Q +,define a vertex-weight function w v : V Q + by setting w v (u) = min e δ(u) {w(e)} for every u V. Let D be a minimum-weight EDS with respect to w, and let C be a minimum-weight vertex cover with respect to w v. Since V (D ) is a vertex cover for G,
WEIGHTED EDGE-DOMINATING SET PROBLEM 321 w v (C ) w v (V (D )). By the construction of w v, for each u V (D ) Hence, w v (u) min e δ(u) D {w(e)}. w v (C ) w v (V (D )) = u V (D ) w v (u) 2w(D ). (1) Suppose we use an r WVC -approximation algorithm to obtain a vertex cover C such that w v (C) r WVC w v (C ). We can construct an edge-dominating set D C from C by selecting a minimum-weight edge in δ(u) for each u C. Thus w(d C ) w v (C). Combining this with (1) we have w(d C ) r WVC w v (C ) 2r WVC w(d ), which establishes the theorem. As mentioned earlier, the smallest value of r WVC currently known for general weighted log log V graphs is 2 (Hochbaum, 1997), yielding an EDS approximation ratio of 4 2 log V log log V. Of course, for special classes we can do better. For instance exact polynomialtime algorithms exist for weighted VC on bipartite graphs, yielding a 2-approximation for log V weighted EDS on bipartite graphs. 4. A 2 1 -approximation: Reduction to edge cover 4.1. Polyhedra Given an instance G = (V, E) and a corresponding cost vector c Q E +, we may formulate the weighted edge-dominating set problem as an integer program min c e x e (EDS(G)) subject to: e E x(δ(u)) + x(δ(v)) x uv 1 {u,v} E x e {0, 1} e E. The constraints of (EDS(G)) ensure that each edge is covered by at least one edge. Relaxing the 0-1 constraints yields min c e x e (FEDS(G)) subject to: e E x(δ(u)) + x(δ(v)) x uv 1 {u,v} E x e 0 e E.
322 CARR ET AL. We henceforth assume without loss of generality that G has no isolated vertices, since deleting such vertices does not affect an edge-dominating set. In our reduction to edge cover we will also be interested in min c e x e (FEC(G)) subject to: e E x(δ(u)) 1 u V x e 0 e E. It is easy to see that the incidence vector of any edge cover for G satisfies all the constraints in (FEC(G)), hence is feasible for it. However, (FEC) may not have integral optimal solutions in general, to which a unit-weighted triangle attests. The optimal solution for (FEC) has x e = 1/2, for all e E, for a total weight of 3/2, while the weight of an integral solution must be at least 2. Thus the inequalities (FEC) are not sufficient to define (EC), the convex hull of the incidence vectors of edge covers. Fortunately, due to a result of Edmonds and Johnson (1970), the complete set of linear inequalities describing (EC) is in fact known. Proposition 3 (Edmonds and Johnson (1970)). The edge cover polytope (EC(G)) can be described by the set of linear inequalities of (FEC(G)) in addition to x(e(s)) + x(δ(s)) S +1 2 S V, S odd. (2) 4.2. Algorithm Let x be a feasible solution for (FEDS(G)). Since for each {u,v} E, x(δ(u)) + x(δ(v)) 1 + x uv,wehavemax{x(δ(u)), x(δ(v))} 1+x uv 1. We use this criterion to define a 2 2 vertex set V + as follows. For each edge {u,v} E we select the endpoint whose fractional degree achieves max{x(δ(u)), x(δ(v))} to be in V + ; in the case of a tie, we choose one endpoint arbitrarily. We let V = V \V +. Proposition 4. V + is a vertex cover of G. Since an edge cover of a vertex cover is an edge-dominating set, we have reduced the problem at hand to that of finding a good edge cover of the set of vertices V +. This is not quite the standard edge cover problem, yet a fair amount is known about it. For instance one can reduce this problem to the maximum weight capacitated b-matching problem (see Grötschel et al., 1988, p. 259). In fact a complete polynomial-time separable linear description of the associated polytope is also known (Pulleyblank, 1995). Rather than trouble ourselves with the technicalities that dealing directly with the V + edge cover problem imposes, we show how to reduce an instance of this problem to a bona fide instance of weighted edge cover. We construct a new instance Ḡ = ( V, Ē) such that there is a one-to-one cost preserving correspondence between V + edge covers in G and edge covers of Ḡ. Recall that V + and V partition V.
WEIGHTED EDGE-DOMINATING SET PROBLEM 323 Let the vertex set V be a copy of V, where v V corresponds to v V. We set V = V V and Ē = E E, where E consists of zero-cost edges, one between each v V its copy v V.Nowif D is an edge cover of Ḡ, then D E must be an edge set of equal cost covering all the vertices in V +. Conversely if D + is an edge set covering all the vertices in V +, then D + E is an edge cover of Ḡ of equal cost, since the edges in E cost nothing. We are now in a position to describe the algorithm, which may be stated quite simply as 1. Compute an optimal solution x for (FED(G)). 2. Compute V +. 3. Compute and output a minimum-weight set of edges D covering V +. The algorithm clearly runs in polynomial time as the most expensive step is solving a compact linear program. Note that steps 2 and 3 may be implemented by the transformation above or by any method the reader fancies; however, the true benefit of the transformation may not be fully apparent until we analyze the approximation guarantee of the algorithm. 4.3. Analysis As before suppose we are given an instance graph G = (V, E) with no isolated vertices and a nonnegative cost vector c. Let x be some feasible fractional solution for (FEDS(G)). Along the lines of the algorithm, suppose we have computed V + and the resulting transformed instance, Ḡ = ( V = V V, Ē = E E ). Let x = (x, 1 E ) Q Ē + ; that is, x corresponds to the augmentation of the fractional edge-dominating set x by E, a zero-cost set of edges. Note that by construction, x is feasible for (FEDS(Ḡ)). Similarly we extend c to c = (c, 0 E ) Q Ē +. Note that we have c x = c x. We may now proceed to show that there is an integral edge cover of Ḡ which does not cost too much more than our fractional edge-dominating solution, x. Theorem 3. The point 21 x is feasible for (EC(Ḡ)). Proof: Let ȳ = 2 x. Suppose u is a vertex in V.Ifu V +,wehave x(δ(u)) 1 2 ; otherwise u V V, and we have x e = 1 for all e E, so in either case ȳ(δ(u)) 1, (3) hence ȳ is feasible for (FEC(Ḡ)). Yet this is not quite good enough as (FEC(Ḡ)) does not have integral extreme points in general, so we extend this by showing that increasing ȳ by a 1 20 fraction places it in (EC(Ḡ)). To accomplish this we use the fact that x is a fractional edge-dominating set of Ḡ, hence ȳ satisfies ȳ(δ(u)) +ȳ(δ(v)) 2 +ȳ uv. (4) Armed with this and the constraints of (FEC(Ḡ)), we proceed to show that 21 20 ȳ also satisfies (2) with respect to Ḡ.
324 CARR ET AL. Suppose S is a subset of V of odd cardinality; let s = S. When s = 1, the constraints (2) are trivially satisfied by ȳ, so suppose s 3. By combining (3) and (4) we see { 2 +ȳuv if uv Ē, ȳ(δ(u)) +ȳ(δ(v)) 2 otherwise. Summing the appropriate inequality above for each pair {u, v} in S S, where u v, we get (s 1)ȳ(δ(S)) + 2(s 1)ȳ(Ē(S)) = (s 1) ȳ(δ(u)) u S = ȳ(δ(u)) +ȳ(δ(v)) {{u,v} S S u v} s(s 1) +ȳ(ē(s)). Isolating the desired left hand side yields ȳ(δ(s)) +ȳ(ē(s)) Using standard optimization techniques, { } s+1 2 max = 21 s 3,odd 20, s(s 1) 2s 3 which is achieved when s = 5. s(s 1) + (s 2)ȳ(δ(S)) 2s 3 s(s 1), for s 3. 2s 3 The theorem implies that 21 x is a convex combination of integral edge covers of Ḡ, which by the construction of Ḡ are also integral edge-dominating sets of Ḡ containing the edges in E. Equivalently the theorem implies that 21 x is a convex combination of integral V + edge covers in G, which since V + is a vertex cover of G, are also integral edge-dominating sets of G. Thus there must be an integral edge-dominating set D of G of cost at most 21 c x. In particular, when x is an optimal fractional edge-dominating set of G and z EDS the cost of an optimal integral solution, we find an integral solution of cost at most 21 c x 21 z EDS. Corollary 1. The point 21 x is feasible for (EDS(G)) when x (FEDS(G)). Corollary 2. The algorithm of Section 4.2 generates a solution of cost at most 2 1 times the optimal. Note that when G is bipartite, Ḡ is bipartite as well. In this case (FEC(G)) forms a totally unimodular constraint set, hence 2 x is feasible for (EC(Ḡ)). Proposition 5. The algorithm of Section 4.2 generates a solution of cost at most 2 times the optimal on a bipartite instance graph. This is in fact asymptotically tight as figure 1 demonstrates.
WEIGHTED EDGE-DOMINATING SET PROBLEM 325 Figure 1. A fractional extreme point of cost k + 1 2. The algorithm chooses the darkened vertices as V +, yielding a solution of cost 2k, while the optimal integral solution costs k + 1. 4.4. An extension Suppose we are given an instance in which we are asked to find a minimum-weight edge set which dominates a specified subset F E. We need only modify our algorithm so that only endpoints of edges in F are considered for V +. The analysis of the previous section remains essentially the same, and our solution will cover the vertices in V + which ensures that the edges in F will be dominated. 4.5. Integrality gap Given that weighted EDS is as hard to approximate as weighted VC and that no polynomialtime algorithm with a constant performance guarantee strictly less than 2 is known for the latter, we might indulge in a respite from developing EDS algorithms if the former were shown to be approximable with a factor of 2. Unfortunately, it turns out that as long as our algorithm analysis is based exclusively on the optimal cost of (FEDS) as a lower bound for that of (EDS), we should relinquish such hope. The formulation (FEDS) introduces an integrality gap, { } minx (EDS(G)) c x max G=(V,E), c Q E min + x (FEDS(G)) c x larger than 2. Corollary 1 bounds it above by 2 1, and it will be shown below that this is in fact a tight bound. Consider the complete graph on 5n vertices, and let G 1,...,G n be n vertex disjoint subgraphs, each isomorphic to K 5. Assign to each edge of G i a weight of 1, and assign to any edge not in any of these subgraphs, some large weight. Let x e = 1/7 ife is an edge of some G i and x e = 0 otherwise. Then it can be verified that x(δ(e)) 1 for all e, hence x is a feasible solution for (FEDS(K 5n )) of cost n. On the other hand, any integral solution 7 must cover all but one vertex in the graph. Prohibited to pick an edge outside of some G i,
326 CARR ET AL. an integral solution of small cost would choose 3 edges from each of G i s but one for a total cost of 3n 1. Thus the integrality gap of formulation (FEDS) approaches 2 1. Although this example establishes the integrality gap of the formulation we employ, our algorithm may still perform provably better. The class of graphs depicted in figure 1 preclude it from guaranteeing a bound less than 2, even for the unweighted bipartite case, and we offer our gratitude for a proof that it is indeed a 2-approximation. The integrality gap of (FEDS(G)) is at most 2 when G is bipartite (Proposition 5); in fact it may grow arbitrarily close to 2. Let G be a complete bipartite graph K n,n with unit weights. Then, x e = 1 2n 1 for all e E is a feasible solution of cost n 2. Any integral solution must 2n 1 contain k edges since it must cover all of the vertices in at least one vertex class of the bipartition, so the integrality gap must be at least n(2n 1) = 2 1 n 2 n. References S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy, Proof verification and hardness of approximation problems, in Proceedings of the 33rdAnnual IEEE Symposium on Foundations of Computer Science, 1992, pp. 14 23. B. Baker, Approximation algorithms for NP-complete problems on planar graphs, J. ACM, vol. 41, pp. 153 180, 1994. V. Chvátal, A greedy heuristic for the set-covering problem, Math. Oper. Res., vol. 4, no. 3, pp. 233 235, 1979. J. Edmonds and E. Johnson, Matching, a well solved class of integer linear programs, in Combinatorial Structures and Their Applications, Gordon & Breach: New York, 1970, pp. 89 92. U. Feige, A threshold of ln n for approximating set cover, in Proceedings of the 28th Annual ACM Symposium on Theory of Computing, May 1996, pp. 314 318. M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, 1988. F. Harary, Graph Theory, Addison-Wesley: Reading, MA, 1969. D.S. Hochbaum (Ed.), Approximation Algorithms for NP-hard Problems, PWS Publishing Company: Boston, MA, 1997. J. Horton and K. Kilakos, Minimum edge dominating sets, SIAM J. Discrete Math., vol. 6, no. 3, pp. 375 387, 1993. H. Hunt III, M. Marathe, V. Radhakrishnan, S. Ravi, D. Rosenkrantz, and R. Stearns, A unified approach to approximation schemes for NP- and PSPACE-hard problems for geometric graphs, in Proc. 2nd Ann. European Symp. on Algorithms, 1994, pp. 424 435. D.S. Johnson, Approximation algorithms for combinatorial problems, J. Comput. System Sci., vol. 9, pp. 256 278, 1974. L. Lovász, On the ratio of optimal integral and fractional covers, Discrete Math., vol. 13, pp. 383 390, 1975. S. Mitchell and S. Hedetniemi, Edge domination in trees, in Proc. 8th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, 1977, pp. 489 509. K.G. Murty and C. Perin, A 1-matching blossom-type algorithm for edge covering problems, Networks, vol. 12, pp. 379 391, 1982. C. Papadimitriou and M. Yannakakis, Optimization, approximation and complexity classes, J. Comput. System Sci., vol. 43, pp. 425 440, 1991. W.R. Pulleyblank, Matchings and extensions, in Handbook of Combinatorics, vol. 1, Elsevier, 1995, pp. 179 232. A. Srinivasan, K. Madhukar, P. Nagavamsi, C.P. Rangan, and M.-S. Chang, Edge domination on bipartite permutation graphs and cotriangulated graphs, Information Processing Letters, vol. 56, pp. 165 171, 1995. M. Yannakakis and F. Gavril, Edge dominating sets in graphs, SIAM J. Appl. Math., vol. 38, no. 3, pp. 364 372, 1980.