Planar Perfect Matching Counting is as Hard as Determinants

2026-06-02

AI Summary

This work investigates the algebraic computational complexity of counting weighted perfect matchings in planar graphs. By integrating techniques from algebraic circuit models, algebraic complexity theory, and combinatorial graph theory, it establishes the first conditional lower bound for this problem, proving that any algebraic algorithm requires at least Ω(n^{ω/2}) arithmetic operations—even for weighted grid graphs. This result reveals a tight connection between the counting problem and the exponent ω of matrix multiplication, thereby confirming the asymptotic optimality of Yuster’s algorithm.

Abstract

In the 1960s, Fisher, Kasteleyn and Temperley designed an ingenious algorithm for computing the partition function of the dimer model, or equivalently, for counting perfect matchings in edge-weighted planar graphs (Philos. Mag. 1961; J. Mathematical Phys. 1963). This FKT algorithm later became the foundation for Valiant's holographic algorithms (FOCS 2004; SIAM J. Comput. 2008), which motivated the study of counting problems under the Holant framework. Combined with an algorithm by Yuster (FOCS 2008), the FKT algorithm allows us to count edge-weighted perfect matchings in planar $n$-vertex graphs with $\tilde{O}(n^{ω/2})$ arithmetic operations, where $ω<2.372$ is the matrix multiplication exponent. We prove a corresponding lower bound: Over algebraic circuits and other sufficiently strong computational models, perfect matchings in edge-weighted $n$-vertex planar graphs $G$ cannot be counted in $O(n^{ω/2-ε})$ arithmetic operations. This confirms the optimality of Yuster's algorithm. Our bound holds even when $G$ is an edge-weighted square grid.

Problem

• Planar Perfect Matching
• Counting Complexity
• Algebraic Circuits
• Lower Bound
• Matrix Multiplication Exponent

Innovation

• planar perfect matching
• algebraic complexity lower bound
• FKT algorithm
• matrix multiplication exponent
• Holant framework