On the [formula omitted]-edge-incident subgraph problem and its variants

Abstract

Given a node-weighted graph G=(V,E) and an integer k, the k-edge-incident subgraph problem requires one to find a vertex set S⊆V of maximum weight that covers at most k edges, and the minimum partial vertex cover problem requires one to find a set of k vertices that covers the minimum number of edges. These two problems are closely related to the well-studied densestk-subgraph problem, and are interesting on their own. In this paper, we study these two problems from an approximation point of view. We obtain the following results. 1.For the k-edge-incident subgraph problem, we present a (2+ε) approximation algorithm for any fixed ε>0, which improves the previous best approximation ratio of 3 and matches that of its unweighted version [O. Goldschmidt and D.S. Hochbaum, k-edge subgraph problems, Discrete Appl. Math. 74 (2) (1997) 159–169].2.For the minimum partial vertex cover problem, we give a 2-approximation algorithm. We then propose a polynomial-time approximation scheme (PTAS) for it on the class of everywhere-c-dense graphs on which many well-studied combinatorial problems have been investigated.