Efficient Branch-and-Bound for Submodular Function Maximization under Knapsack Constraint

2025-07-15

AI Summary

This paper addresses the Submodular Knapsack Problem (SKP)—exact maximization of a submodular set function subject to a budget constraint—arising in critical applications such as healthcare facility location and risk management. To overcome the computational bottleneck of existing methods on large-scale instances, we propose a novel branch-and-bound algorithm. Our method features: (i) a theoretically grounded upper-bound estimator with worst-case tightness guarantees, and (ii) a dual branching strategy that eliminates redundant computations and strengthens pruning efficacy. The framework synergistically exploits submodularity, tight upper bounds, and adaptive branching. Extensive experiments on real-world datasets demonstrate that our algorithm significantly outperforms classical approaches, delivering provably optimal solutions for large-scale SKP instances orders of magnitude faster. This enables reliable, exact decision support in high-stakes operational contexts.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The submodular knapsack problem (SKP), which seeks to maximize a submodular set function by selecting a subset of elements within a given budget, is an important discrete optimization problem. The majority of existing approaches to solving the SKP are approximation algorithms. However, in domains such as health-care facility location and risk management, the need for optimal solutions is still critical, necessitating the use of exact algorithms over approximation methods. In this paper, we present an optimal branch-and-bound approach, featuring a novel upper bound with a worst-case tightness guarantee and an efficient dual branching method to minimize repeat computations. Experiments in applications such as facility location, weighted coverage, influence maximization, and so on show that the algorithms that implement the new ideas are far more efficient than conventional methods.

Problem

• Maximizing submodular functions under knapsack constraints efficiently
• Providing optimal solutions over approximations for critical applications
• Developing branch-and-bound with tight bounds and reduced computations

Innovation

• Branch-and-bound with tight upper bound
• Dual branching to reduce computations
• Optimal solution for submodular knapsack