On Approximability of the Independent/Connected Edge Dominating Set Problems

Information Processing Letters 79 (2001) 261–266 On approximability of the independent/connected edge dominating set problems Toshihiro Fujito Department of Electronics, Nagoya University, Furo, Chikusa, Nagoya, 464-8603 Japan Received 16 June 2000; received in revised form 5 December 2000 Communicated by K. Iwama Abstract We investigate polynomial-time approximability of the problems related to edge dominating sets of graphs. When edges are unit-weighted, the edge dominating set problem is polynomially equivalent to the minimum maximal matching problem, in either exact or approximate computation, and the former problem was recently found to be approximable within a factor of 2 even with arbitrary weights. It will be shown, in contrast with this, that the minimum weight maximal matching problem cannot be approximated within any polynomially computable factor unless P = NP. The connected edge dominating set problem and the connected vertex cover problem also have the same approximability when edges/vertices are unit-weighted. The former problem is already known to be approximable, even with general edge weights, within a factor of 3.55. We will show that, when general weights are allowed, (1) the connected edge dominating set problem can be approximated within a factor of 3 + ε, and (2) the connected vertex cover problem is approximable within a factor of ln n + 3 but cannot be within (1 − ε)ln n for any ε>0 unless NP ⊂ DTIME(nO(loglogn)). 2001 Elsevier Science B.V. All rights reserved. Keywords: Combinatorial optimization problems; Approximation algorithms; Edge dominating set 1. Introduction edges in a graph. The problem EDS is then that of find- ing a smallest eds or, if edges are weighted, an eds of In this paper we investigate polynomial-time ap- minimum total weight. Yannakakis and Gavril showed proximability of problems related to edge dominating that EDS is NP-complete even when the graphs are sets of graphs. For two pairs of problems considered, planar or bipartite of maximum degree 3 [20]. A set it will be shown that, while both problems in each of edges is called a matching (or independent)ifno pair have the same approximability for the unweighted two of them have a vertex in common, and a match- case, their approximation properties differ drastically ing is maximal if no other matching properly con- when general non-negative weights are allowed. tains it. Notice that any maximal matching is neces- In an undirected graph an edge dominates all the sarily an eds, because an edge not in it must be ad- edges adjacent to it, and an edge dominating set (eds) jacent to some in it. For this reason it is also called is a set of edges collectively dominating all the other an independent edge dominating set, and the problem IEDS asks for a minimum maximal matching in E-mail address: [email protected] (T. Fujito). a given graph. Interestingly, one can also construct a 0020-0190/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII:S0020-0190(01)00138-7 262 T. Fujito / Information Processing Letters 79 (2001) 261–266 maximal matching, from any eds, of no larger size in After improving this bound to 3 + ε, we will show that polynomial time [11], implying that EDS and IEDS weighted CVC is as hard to approximate as weighted are polynomially equivalent, in exact or approximate set cover, indicating that it is not approximable within computation, when graphs are unweighted. Based on a factor better than (1 − ε)lnn unless this and the fact that any maximal matching cannot NP ⊂ DTIME nO(loglogn) be more than twice larger than any maximal matching, it has been long known that either problem, with- [4]. Lastly, we present an algorithm approximating out weights, can be approximated within a factor of 2. weighted CVC within a factor of Moreover, weighted EDS was recently shown approx- H(∆− 1) + r ln(∆ − 1) + 3, imable within a factor of 2 [5,3]. In contrast with this, wvc we will present strong inapproximability results for where H(k) is the kth harmonic number and ∆ is the weighted IEDS. maximal vertex degree of a graph. A related problem, We next consider EDS with connectivity require- connected dominating set, can be approximated within ment, called the connected edge dominating set a factor of ln n + 3 for the unweighted case [8], but (CEDS) problem, where it is asked to compute a con- when weighted, the best known bound is 1.35 ln n [9]. nected eds (ceds) of minimum weight in a given connected graph. Since it is always redundant to form a cycle in a ceds, the problem can be restated as that of 2. Independent edge dominating set going after a minimum tree whose vertices “cover” all the edges in a graph, and thus, it is also called tree To show that it is extremely hard to approximate cover.Thevertex cover (VC) problem is another ba- IEDS, we use a reduction from 3SAT. Let us first sic NP-complete graph problem [13], in which a min- describe a general construction of graph Gφ for a imum vertex set is sought in G s.t. every edge of G given 3SAT instance (i.e., a CNF formula) φ,by is incident to some vertex in the set. When a vertex adapting the one used in reducing SAT to minimum cover is additionally required to induce a connected maximal independent set [12,10] to our case. For subgraph in a given connected graph, the problem is simplicity, every clause of φ is assumed without loss called connected vertex cover (CVC) and known to be of generality to contain exactly three literals. Each as hard to approximate as VC is [6]. variable xi appearing in φ is represented in Gφ by These problems are closely related to EDS and two edges adjacent to each other, and the endvertices CEDS in that an edge set F is an eds for G iff of such a path of length 2 are labeled xi and x¯i;let V(F), the set of vertices touched by edges of F ,is Ev denote the set of these edges. Each clause cj of φ a vertex cover for G, and similarly, a tree F is a is represented by a triangle (a cycle of length 3) Cj , ceds iff V(F) is a connected vertex cover. It follows and vertices of Cj are labeled distinctively by literals that since the number of vertices is that of edges appearing in cj ;letEc denote the set of edges in these plus one in any tree, the unweighted versions of disjoint triangles. The paths in Ev and triangles in Ec CEDS and CVC have the same approximability, and are connected together by having an edge between in fact they are known to be approximable within a every vertex of each triangle and the endvertex of a factor of 2 [18,1]. It is not known, however, if CEDS path having the same label. The set of these edges and CVC can be somehow related even if general lying between Ev and Ec is denoted by Eb.Itisa weights are allowed. An algorithm scheme of Arkin simple matter to verify that, for a 3SAT instance φ et al. for weighted CEDS gives its approximation with m variables and p clauses, Gφ constructed this factor in the form of rSt + rwvc(1 + 1/k),forany way consists of 3(m + p) vertices and 2(m + 3p) constant k,whererSt (rwvc) is the performance ratio edges. of any polynomial time algorithm for the Steiner tree (weighted vertex cover, respectively) problem [1]. By Lemma 1. Let M(G) denote a minimum maximal using the currently best algorithms for Steiner tree [17] matching M in G. For any 3SAT instance φ with m and for weighted vertex cover [2] in their scheme, variables and p clauses, and for any number t,there the bound for weighted CEDS is estimated at 3.55. exists a graph Gφ on 3(m+p) vertices and 2(m+3p) T. Fujito / Information Processing Letters 79 (2001) 261–266 263 edges, and a weight assignment w : E →{1,t},such 3. Connected edge dominating set that We first consider a restricted version of CEDS; for m + p, if φ is satisfiable, w M(Gφ) a designated vertex r called root,anr-ceds is a ceds >t, otherwise. touching r, and the problem r-CEDS is to compute an r-ceds of minimum weight. Given an undirected = ∈ ∪ = Proof. Let w(e) 1ife Ev Ec and w(e) t if graph G = (V, E) with edge weights w : E → Q+, e ∈ Eb. Suppose that φ is satisfiable, and let τ be a let G = (V, E) denote its directed version obtained particular truth assignment satisfying φ. Construct a by replacing each edge {u, v} of G by two directed matching Mτ in Ev by choosing, for each i, the edge ones, (u, v) and (v, u), each of weight w({u, v}).For with its endvertex labeled by xi if τ(xi) is true and the root r, a non-empty set S ⊆ V −{r} is called the one having an endvertex labeled by x¯i if τ(xi) is dependent if S is not an independent set in G. Suppose false. Consider any triangle Cj in Ec.Sinceτ satisfies T ⊆ E is an r-ceds, and let T denote the directed φ, at least one edge among those in Eb connecting counterpart obtained by choosing, for each pair of Cj and Ev must be dominated by Mτ .

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