We study the generalized min sum set cover (GMSSC) problem, wherein given a collection of hyperedges E with arbitrary covering requirements {ke ∈ Z+ : e ∈ E}, the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a hyperedge e is considered covered by the first time when ke many of its vertices appear in the ordering. We give a 4.642 approximation algorithm for GMSSC, coming close to the best possible bound of 4, already for the classical special case (with all ke = 1) of min sum set cover (MSSC) studied by Feige, Lovász and Tetali , and improving upon the previous best known bound of 12.4 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. This also gives an LP-based 4 approximation for MSSC. 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. Another well-known special case is the min sum vertex cover (MSVC) problem, in which the input hypergraph is a graph (i.e., |e| = 2) and ke = 1, for every edge e ∈ E. We give a 16/9 ' 1.778 approximation for MSVC, and show a matching integrality gap for the natural LP relaxation. This improves upon the previous best 1.999946 approximation of Barenholz, Feige and Peleg . (The claimed 1.79 approximation result of Iwata, Tetali and Tripathi  for the MSVC turned out have an unfortunate, seemingly unfixable, mistake in it.) Finally, we revisit MSSC and consider the lp norm of cover-time of the hyperedges. Using a dual fitting argument, we show that the natural greedy algorithm simultaneously achieves approximation guarantees of (p + 1)1+1/p, for all p ≥ 1, giving another proof of the result of Golovin, Gupta, Kumar and Tangwongsan , and showing its tightness up to NP-hardness. For p = 1, this gives yet another proof of the 4 approximation for MSSC.

doi.org/10.1137/1.9781611976465.62
32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Bansal, N, Batra, J, Farhadi, M, & Tetali, P. (2021). Improved approximations for min sum vertex cover and generalized min sum set cover. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 986–1005). doi:10.1137/1.9781611976465.62