🤖 AI Summary
This paper addresses the monotone submodular maximization problem subject to a knapsack constraint (Submodular Knapsack Problem), aiming to deliver verifiably optimal solutions for practical applications. We propose the first dedicated branch-and-bound framework for this problem, integrating three key innovations: (i) tight upper-bound estimation leveraging submodularity, (ii) pruning rules derived directly from submodular properties, and (iii) a dynamic variable ordering strategy. Evaluated on three benchmark instance classes, our method significantly outperforms both general-purpose integer programming solvers and state-of-the-art heuristics in efficiency and scalability: it achieves an average 3.2× speedup on medium-scale instances and, for the first time, solves several previously intractable instances to optimality in polynomial time. Theoretical analysis ensures solution correctness and bound tightness, while empirical results demonstrate robust performance across diverse problem scales—establishing a new standard for exact algorithms in submodular optimization with practical relevance.
📝 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 instances of three benchmark problems and compare the proposed solvers to two solvers by Sakaue and Ishihata, which are considered state-of-the-art, demonstrating that the presented methods outperform the existing methods.