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We consider the problem of solving the Min-Sum Submodular Cover problem usinglocal search. The Min-Sum Submodular Cover problem generalizes the NP-completeMin-Sum Set Cover problem, replacing the input set cover instance with amonotone submodular set function. A simple greedy algorithm achieves anapproximation factor of 4, which is tight unless P=NP [Streeter and Golovin,NeurIPS, 2008]. We complement the greedy algorithm with analysis of a localsearch algorithm. Building on work of Munagala et al. [ICDT, 2005], we showthat, using simple initialization, a straightforward local search algorithmachieves a $(4+\epsilon)$-approximate solution in time$O(n^3\log(n/\epsilon))$, provided that the monotone submodular set function isalso second-order supermodular. Second-order supermodularity has been shown tohold for a number of submodular functions of practical interest, includingfunctions associated with set cover, matching, and facility location. Wepresent experiments on two special cases of Min-Sum Submodular Cover and findthat the local search algorithm can outperform the greedy algorithm on smalldata sets.

A Local Search Algorithm for the Min-Sum Submodular Cover Problem