An Approximation Algorithm for Monotone Submodular Cost Allocation

In this paper, we consider the minimum submodular cost allocation (MSCA) problem. The input of MSCA is $k$ non-negative submodular functions $f_1,\ldots,f_k$ on the ground set $N$ given by evaluation oracles, and the goal is to partition $N$ into $k$ (possibly empty) sets $X_1,\ldots,X_k$ so that $\sum_{i=1}^k f_i(X_i)$ is minimized. In this paper, we focus on the case when $f_1,\ldots,f_k$ are monotone (denoted by Mono-MSCA). We provide a natural LP-relaxation for Mono-MSCA, which is equivalent to the convex program relaxation introduced by Chekuri and Ene. We show that the integrality gap of the LP-relaxation is at most $k/2$, which yields a $k/2$-approximation algorithm for Mono-MSCA. We also show that the integrality gap of the LP-relaxation is at least $k/2-\epsilon$ for any constant $\epsilon>0$ when $k$ is fixed.