Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs

Bruno Escoffier, Laurent Gourvès, Jérôme Monnot
2010 Journal of Discrete Algorithms  
We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). Finally, dealing with hypergraphs, we study the complexity and the approximability of two natural generalizations. 37
more » ... us related works The main complexity and approximability results known on this problem are the following: in [34] , it is shown that MinCVC is polynomially solvable when the maximum degree of the input graph is at most 3. However, it is NP-hard in planar bipartite graphs of maximum degree 4 [14] , in planar biconnected graphs of maximum degree 4 [30], as well as in 3-connected graphs [35] . Concerning the positive and negative results of the approximability of this problem, MinCVC is 2-approximable in general graphs [31,2] but it is NP-hard to approximate within ratio 10 . In [11] , the authors propose a general technique to derive PTASs for bidimensional problems. In particular, they give a quasi-PTAS (a (1 + ε)approximation algorithm with running time almost polynomial n O (log log n) when ε is fixed) for the minimum connected dominating set in a class of graphs that includes planar graphs and claim that an analogous result holds for MinCVC. The minimum connected dominating set problem asks to find a dominating set of of minimum size and where the subgraph of G induced by S is connected. Finally, recently the fixed-parameter tractability of MinCVC with respect to the vertex cover size or to the treewidth of the input graph has been studied in [14,21,26-28]. More precisely, in [14] a parameterized algorithm for MinCVC with complexity O * (2.9316 k ) is presented improving the previous algorithm with complexity O * (6 k ) given in [21] where k is the size of an optimal connected vertex cover. Independently, the authors of [26,27] have also obtained FPT algorithms for MinCVC and they obtain in [27] an algorithm with complexity O * ( 2.7606 k ) . In [28] , the author gives a parameterized algorithm for MinCVC with complexity O * (2 t · t 3t+2 n) where t is the treewidth of the graph and n the number of vertices. MinCVC is related to the unweighted version of tree cover. The tree cover problem was introduced in [2] and consists, given a connected graph G = (V , E) with non-negative weights w on the edges, in finding a tree T = (S, E ) of G with S ⊆ V and E ⊆ E satisfying ∀{x, y} ∈ E \ E , {x, y} ∩ S = ∅ and such that w(T ) = e∈E w(e) is minimum. In [2], the authors prove that the tree cover problem is approximable within factor 3.55 and the unweighted version is 2-approximable. Recently, (weighted) tree cover has been shown to be approximable within a factor of 3 in [25] , and a 2-approximation algorithm is proposed in [16] . Clearly, the unweighted version of tree cover is (asymptotically) equivalent to the connected vertex cover. Actually, from any connected vertex cover S of G, let T = (S, E ) be any spanning tree of G[S], the subgraph of G induced by S; T = (S, E ) is a tree cover of G with weight |S| − 1. Conversely, if T = (S, E ) is a tree cover of G, then S is a connected vertex cover of G (of size |E | + 1). Our contribution In this article, we mainly deal with complexity and approximability issues for MinCVC in particular classes of graphs. More precisely,
doi:10.1016/j.jda.2009.01.005 fatcat:sekds3g7vvcb3hgynjppurrgge