In this paper, we consider the minimum submodular cost allocation (MSCA) problem. The input of MSCA is $k$ non-negative submodular functions $f_1,f_2,\ldots,f_k$ on the ground set $N$ given by evaluation oracles, and the goal is to partition $N$ into $k$ (possibly empty) sets $S_1,S_2,\ldots,S_k$ so that $\sum_{i=1}^k f_i(S_i)$ is minimized. In this paper, we focus on the case when $f_1,f_2,\ldots,f_k$ are monotone, which coincides with the submodular facility location problem considered by Svitkina and Tardos. We show that the integrality gap of a natural LP-relaxation for MSCA with monotone submodular functions is at most $k/2$, yielding a $k/2$-approximation algorithm. We also prove a nearly matching lower bound: the integrality gap is at least $k/2-ε$ for any constant $ε>0$ when $k$ is fixed.

翻译：本文研究最小次模成本分配（MSCA）问题。MSCA问题的输入为定义在基础集$N$上的$k$个非负次模函数$f_1,f_2,\ldots,f_k$（通过求值预言机给出），目标是将$N$划分为$k$个（可能为空的）集合$S_1,S_2,\ldots,S_k$，以最小化$\sum_{i=1}^k f_i(S_i)$。本文重点研究$f_1,f_2,\ldots,f_k$具有单调性的情形，该情形与Svitkina和Tardos研究的次模设施选址问题相一致。我们证明：对于单调次模函数的MSCA问题，其自然线性规划松弛的整数间隙至多为$k/2$，从而得到一个$k/2$近似算法。同时我们给出了近乎匹配的下界：当$k$固定时，对于任意常数$ε>0$，整数间隙至少为$k/2-ε$。