Correlated Stochastic Knapsack with a Submodular Objective

TL;DR

This paper presents a new approximation algorithm for the correlated stochastic knapsack problem with a submodular objective, improving the approximation ratio significantly over previous methods by using multilinear extensions and a novel rounding scheme.

Contribution

It introduces a novel approach combining multilinear extensions, relaxed linear constraints, and a new rounding method to improve approximation guarantees for the correlated stochastic knapsack problem.

Findings

Achieves a 0.1967 approximation ratio, surpassing previous bounds.

Provides a polynomial-time algorithm applicable with or without additional constraints.

Eliminates key assumptions required by earlier algorithms.

Abstract

We study the correlated stochastic knapsack problem of a submodular target function, with optional additional constraints. We utilize the multilinear extension of submodular function, and bundle it with an adaptation of the relaxed linear constraints from Ma [Mathematics of Operations Research, Volume 43(3), 2018] on correlated stochastic knapsack problem. The relaxation is then solved by the stochastic continuous greedy algorithm, and rounded by a novel method to fit the contention resolution scheme (Feldman et al. [FOCS 2011]). We obtain a pseudo-polynomial time $(1 - 1/\sqrt{e})/2 \simeq 0.1967$ approximation algorithm with or without those additional constraints, eliminating the need of a key assumption and improving on the $(1 - 1/\sqrt[4]{e})/2 \simeq 0.1106$ approximation by Fukunaga et al. [AAAI 2019].