Monotone Circuit Complexity of Matching

2025-07-21

AI Summary

This paper establishes an exponential lower bound on the monotone circuit complexity of the perfect matching function on $n$-vertex graphs. Prior to this work, the best known lower bound was Razborov’s quasi-polynomial bound of $n^{Omega(log n)}$ (1985). To overcome this barrier, we introduce a novel sunflower lemma tailored to the combinatorial structure of matchings and integrate it with the standard approximation method for monotone circuits. Our lemma exposes an intrinsic limitation of monotone circuits in capturing matching properties, thereby yielding a stronger combinatorial barrier. As a result, we prove that any monotone circuit computing the perfect matching function must have size at least $2^{n^{Omega(1)}}$. This constitutes the first substantial improvement over Razborov’s bound in nearly four decades, marking a breakthrough from quasi-polynomial to exponential lower bounds for this fundamental problem in circuit complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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
• Security and Privacy: Large-scale security measurements

Abstract

We show that the perfect matching function on $n$-vertex graphs requires monotone circuits of size $smash{2^{n^{Ω(1)}}}$. This improves on the $n^{Ω(log n)}$ lower bound of Razborov (1985). Our proof uses the standard approximation method together with a new sunflower lemma for matchings.

Problem

• Determine monotone circuit size for perfect matching
• Improve Razborov's lower bound for matching complexity
• Introduce new sunflower lemma for matching proofs

Innovation

• Monotone circuits analyze perfect matching complexity
• Exponential size lower bound for matching circuits
• New sunflower lemma enhances approximation method