A Tight Deterministic Algorithm for the Submodular Multiple Knapsack Problem

Submodular function maximization has been a central topic in the theoretical computer science community over the last decade. Plenty of well-performing approximation algorithms have been designed for the maximization of (monotone or non-monotone) submodular functions over a variety of constraints. In this paper, we consider the submodular multiple knapsack problem (SMKP), which is the submodular version of the well-studied multiple knapsack problem (MKP). Roughly speaking, the problem asks to maximize a monotone submodular function over multiple bins (knapsacks). Recently, Fairstein et al. (ESA20) presented a tight $(1-1/e-\epsilon)$-approximation randomized algorithm for SMKP. Their algorithm is based on the continuous greedy technique which inherently involves randomness. However, the deterministic algorithm of this problem has not been understood very well previously. In this paper, we present a tight $(1-1/e-\epsilon)$ deterministic algorithm for SMKP. Our algorithm is based on reducing SMKP to an exponential-size submodular maximizaion problem over a special partition matroid which enjoys a tight deterministic algorithm. We develop several techniques to mimic the algorithm, leading to a tight deterministic approximation for SMKP.