On the Complexity of the Odd-Red Bipartite Perfect Matching Polytope

2026-03-18

AI Summary

This study investigates the description complexity of the perfect matching polytope in bipartite graphs constrained to contain an odd number of red edges. By integrating combinatorial polyhedral analysis, extended formulation complexity, and hardness results for the separation problem, the work establishes that this polytope exhibits an intrinsically complex facial structure: any exact linear description necessarily involves a large number of constraints with coefficients outside {0, ±1}. These findings refute the exactness of existing exponential-size relaxation schemes, demonstrate that the associated separation problem is NP-hard, and extend to bimodular integer programming polyhedra. Consequently, the results confirm that such problems cannot be effectively captured by conventional relaxations relying solely on simple-coefficient inequalities.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The odd-red bipartite perfect matching problem asks to find a perfect matching containing an odd number of red edges in a given red-blue edge-colored bipartite graph. While this problem lies in $\mathsf{P}$, its polyhedral structure remains elusive, despite renewed attention to achieving better polyhedral understanding, nurtured by recent advances from two complementary angles. Apart from being a special case of bimodular integer programs, whose polyhedral structure is also badly understood, it is related to one of the most notorious open derandomization questions in theoretical computer science: whether there is a deterministic efficient algorithm for the exact bipartite perfect matching problem, which asks to find a perfect matching with exactly $k$ red edges. Recent progress towards deterministic algorithms for this problem crucially relies on a good polyhedral understanding. Motivated by this, Jia, Svensson, and Yuan show that the extension complexity of the exact bipartite perfect matching polytope is exponential in general. Interestingly, their result is true even for the easier odd-red bipartite perfect matching problem. For this problem, they introduce an exponential-size relaxation and leave open whether it is an exact description. Apart from showing that this description is not exact and even hard to separate over, we show, more importantly, that the red-odd bipartite perfect matching polytope exhibits complex facet structure: any exact description needs constraints with large and diverse coefficients. This rules out classical relaxations based on constraints with all coefficients in $\{0,\pm1\}$, such as the above-mentioned one, and suggests that significant deviations from prior approaches may be needed to obtain an exact description. More generally, we obtain that also polytopes corresponding to bimodular integer programs have complex facet structure.

Problem

• odd-red bipartite perfect matching
• polyhedral structure
• extension complexity
• bimodular integer programs
• facet complexity

Innovation

• polyhedral complexity
• bimodular integer programming
• odd-red bipartite matching
• facet structure
• extension complexity