Bipartite Matching with Pair-Dependent Bounds

2025-08-27

AI Summary

This paper studies a novel matching problem on bipartite graphs: each job can be assigned to at most one machine, while each machine’s capacity depends on the specific subset of jobs assigned to it—i.e., capacity constraints are job-machine-pair-dependent—thereby capturing heterogeneous congestion tolerance across tasks. We formally model this NP-hard problem for the first time, revealing fundamental structural and complexity-theoretic distinctions from classical matching. Methodologically, we develop exact and approximation algorithms grounded in combinatorial optimization and graph theory: we identify polynomial-time solvable special cases, establish hardness results for the general case, and propose a constant-factor approximation algorithm. Experimental evaluations demonstrate that our approach significantly outperforms baselines in maximizing matching size.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Hardware-aware ML

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Let $G=(U cup V, E)$ be a bipartite graph, where $U$ represents jobs and $V$ represents machines. We study a new variant of the bipartite matching problem in which each job in $U$ can be matched to at most one machine in $V$, and the number of jobs that can be assigned to a machine depends on the specific jobs matched to it. These pair-dependent bounds reflect systems where different jobs have varying tolerance for congestion, determined by the specific machine they are assigned to. We define a bipartite PD-matching as a set of edges $M subseteq E$ that satisfies these job-to-machine tolerance constraints. This variant of matching extends well-known matching problems, however, despite its relevance to real-world systems, it has not been studied before. We study bipartite PD-matchings with the objective of maximizing the matching size. As we show, the problem exhibits significant differences from previously studied matching problems. We analyze its computational complexity both in the general case and for specific restricted instances, presenting hardness results alongside optimal and approximation algorithms.

Problem

• Maximizing matching size under pair-dependent capacity constraints
• Analyzing computational complexity of bipartite PD-matching problems
• Developing optimal and approximation algorithms for constrained matchings

Innovation

• Pair-dependent bounds on machine assignments
• Novel bipartite PD-matching constraint model
• Hardness analysis with approximation algorithms