On Min Sum Vertex Cover and Generalized Min Sum Set Cover

Abstract

We study the Generalized Min Sum Set Cover (GMSSC) problem, wherein given a collection of hyperedges with arbitrary covering requirements , the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a hyperedge is considered covered by the first time when and many of its vertices appear in the ordering. We give a approximation algorithm for GMSSC, coming close to the best possible bound of 4, already for the classical special case (with all ) of Min Sum Set Cover (MSSC) studied by Feige, Lovász, and Tetali, and improving upon the previous best known bound of due to Im, Sviridenko, and van der Zwaan. Our algorithm is based on transforming the LP solution by a suitable kernel and applying randomized rounding. As part of the analysis of our algorithm, we also derive an inequality on the lower tail of a sum of independent Bernoulli random variables, which might be of independent interest and broader utility. Min Sum Vertex Cover (MSVC) is a well-known special case of MSSC in which the input hypergraph is a graph (i.e., ) and for every edge . We give a approximation for MSVC and show a matching integrality gap for the natural LP relaxation. This improves upon the previous best approximation of Barenholz, Feige, and Peleg. Finally, we revisit MSSC and consider the norm of cover-time of the hyperedges. Using a dual fitting argument, we show that the natural greedy algorithm achieves tight, up to NP-hardness, approximation guarantees of for all , giving another proof of the result of Golovin, Gupta, Kumar, and Tangwongsan, and showing its tightness up to NP-hardness. For , this gives yet another proof of the 4 approximation for MSSC.