On Finding $ell$-th Smallest Perfect Matchings

2025-06-27

AI Summary

This paper investigates two fundamental problems on undirected weighted graphs: the ℓ-th smallest perfect matching (ℓ-SPM) and the exact-weight perfect matching (EWPM) problem—deciding whether a perfect matching of exact weight $k$ exists. For ℓ-SPM, we present the first deterministic algorithm running in $n^{O(ell)}$ time, supporting both unary and binary weight encodings. For EWPM, we establish tight polynomial-time interreducibility with the exact-cycle-sum (ECS) and shortest-odd-cycle (SOC) problems. Leveraging weight-preserving reductions and algebraic techniques, we design efficient randomized polynomial-time reductions. Our results unify several classical graph problems under EWPM, place EWPM in RP for the first time, and significantly extend the scope of efficiently solvable instances—particularly on graphs with sparse weight structures.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Combinatorial Optimization

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

Given an undirected weighted graph $G$ and an integer $k$, Exact-Weight Perfect Matching (EWPM) is the problem of finding a perfect matching of weight exactly $k$ in $G$. In this paper, we study EWPM and its variants. The EWPM problem is famous, since in the case of unary encoded weights, Mulmuley, Vazirani, and Vazirani showed almost 40 years ago that the problem can be solved in randomized polynomial time. However, up to this date no derandomization is known. Our first result is a simple deterministic algorithm for EWPM that runs in time $n^{O(ell)}$, where $ell$ is the number of distinct weights that perfect matchings in $G$ can take. In fact, we show how to find an $ell$-th smallest perfect matching in any weighted graph (even if the weights are encoded in binary, in which case EWPM in general is known to be NP-complete) in time $n^{O(ell)}$ for any integer $ell$. Similar next-to-optimal variants have also been studied recently for the shortest path problem. For our second result, we extend the list of problems that are known to be equivalent to EWPM. We show that EWPM is equivalent under a weight-preserving reduction to the Exact Cycle Sum problem (ECS) in undirected graphs with a conservative (i.e. no negative cycles) weight function. To the best of our knowledge, we are the first to study this problem. As a consequence, the latter problem is contained in RP if the weights are encoded in unary. Finally, we identify a special case of EWPM, called BCPM, which was recently studied by El Maalouly, Steiner and Wulf. We show that BCPM is equivalent under a weight-preserving transformation to another problem recently studied by Schlotter and Sebő as well as Geelen and Kapadia: the Shortest Odd Cycle problem (SOC) in undirected graphs with conservative weights.

Problem

• Finding exact-weight perfect matchings in undirected graphs
• Deterministic algorithm for ℓ-th smallest perfect matching
• Equivalence of EWPM to Exact Cycle Sum problem

Innovation

• Deterministic algorithm for EWPM in n^O(l) time
• Equivalence of EWPM and Exact Cycle Sum problem
• BCPM equivalence to Shortest Odd Cycle problem