Improved Budgeted Connected Domination and Budgeted Edge-Vertex Domination
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We consider the budgeted version of the classical connected dominating set problem (BCDS). Given a graph G and an integer budget k, we seek to find a connected subset of at most k vertices which maximizes the number of dominated vertices in G. We answer an open question in [Khuller, Purohit, and Sarpatwar, SODA 2014] and thus we improve over the previous (1-1/e)/13 approximation. Our algorithm provides a (1-1/e)/7 approximation guarantee by employing an improved method for enforcing connectivity and performing tree decompositions. We also consider the edge-vertex domination variant, where an edge dominates its endpoints and all vertices neighboring them. In budgeted edge-vertex domination (BEVD), we are given a graph G, and a budget k, and we seek to find a, not necessarily connected, subset of edges such that the number of dominated vertices in G is maximized. We prove there exists a (1-1/e)-approximation algorithm. Also, for any ϵ > 0, we present a (1-1/e+ϵ)-inapproximability result by a gap-preserving reduction from the maximum coverage problem. We notice that, in the connected case, BEVD becomes equivalent to BCDS. Moreover, we examine the "dual" partial edge-vertex domination (PEVD) problem, where a graph G and a quota n' are given. The goal is to select a minimum-size set of edges to dominate at least n' vertices in G. In this case, we present a H(n')-approximation algorithm by a reduction to the partial cover problem.
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