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

翻译：我们研究使用局部搜索求解最小和子模覆盖（Min-Sum Submodular Cover）问题。该问题推广了NP完全的基于集合覆盖的最小和求解问题，将其中的输入集合覆盖实例替换为单调子模集函数。一种简单的贪心算法可实现近似比为4的求解，该结果在P≠NP假设下是紧的[Streeter与Golovin, NeurIPS, 2008]。我们通过分析局部搜索算法对贪心算法进行补充。基于Munagala等人[ICDT, 2005]的工作，我们证明：通过简单初始化，当单调子模集函数同时满足二阶超模性时，直接采用局部搜索算法可在$O(n^3\log(n/\epsilon))$时间内获得$(4+\epsilon)$近似解。已有研究表明，二阶超模性在众多具有实际意义的子模函数中成立，包括与集合覆盖、匹配及设施选址相关的函数。我们在最小和子模覆盖的两个特例上进行了实验，发现局部搜索算法在小规模数据集上的表现可优于贪心算法。