An Approximation Algorithm for Monotone Submodular Cost Allocation
Ryuhei Mizutani
TL;DR
This work studies the minimum submodular cost allocation problem in the monotone setting (Mono-MSCA) and introduces the LP-Relaxation $\textsc{LP-Rel}$, equivalent to the convex $\textsc{LE-Rel}$ of Chekuri and Ene. The main result shows the integrality gap of $\textsc{LP-Rel}$ is at most $\frac{k}{2}$, yielding a $\frac{k}{2}$-approximation for Mono-MSCA; the authors provide a constructive, polynomial-time method to convert an optimal LP solution into an integral partition using chain supports and carefully selected sets $U_i^j$. They also establish a matching lower bound of $\frac{k}{2}-\varepsilon$ (for fixed $k$) when $\log n \ge k$, demonstrating near-tightness for small $k$; for $k=2$ the result recovers total dual integrality of the unweighted polymatroid intersection. Overall, the paper advances the understanding of approximation guarantees and integrality gaps for submodular cost allocation with small $k$, bridging LP relaxations, Lovász extensions, and polymatroid concepts to yield practical algorithms.
Abstract
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 $k/2-ε$ for any constant $ε>0$ when $k$ is fixed.