An Exact Solver for Submodular Knapsack Problems

TL;DR

This paper introduces an exact branch-and-bound algorithm with acceleration techniques for solving the NP-hard submodular knapsack problem, providing precise solutions where approximations are inadequate.

Contribution

It presents a novel exact solver specifically designed for the submodular knapsack problem, outperforming existing methods in efficiency and solution quality.

Findings

The proposed algorithm achieves high success on benchmark and real-world instances.

Acceleration techniques significantly improve computational efficiency.

Compared to existing solvers, it provides more accurate solutions within reasonable time.

Abstract

We study the problem of maximizing a monotone increasing submodular function over a set of weighted elements subject to a knapsack constraint. Although this problem is NP-hard, many applications require exact solutions, as approximate solutions are often insufficient in practice. To address this need, we propose an exact branch-and-bound algorithm tailored for the submodular knapsack problem and introduce several acceleration techniques to enhance its efficiency. We evaluate these techniques on artificial instances of three benchmark problems as well as on instances derived from real-world data. We compare the proposed solver with two solvers by Sakaue and Ishihata (2018), which currently achieve the strongest performance reported in the literature, as well as with a branch-and-cut algorithm implemented using Gurobi that solves a binary linear reformulation of the submodular knapsack problem, demonstrating that our methods are highly successful.