On Finding $\ell$-th Smallest Perfect Matchings

TL;DR

This paper presents a deterministic algorithm for finding the l-th smallest perfect matching in weighted graphs, establishes its equivalence to the Exact Cycle Sum problem, and explores its relation to the Shortest Odd Cycle problem.

Contribution

It introduces a deterministic l^{O(1)} algorithm for l-th smallest perfect matchings and links EWPM to the Exact Cycle Sum problem and Shortest Odd Cycle problem.

Findings

Deterministic algorithm for l-th smallest perfect matching in time n^{O(l)}

EWPM is equivalent to the Exact Cycle Sum problem under weight-preserving reductions

Identifies special cases linking EWPM to the Shortest Odd Cycle problem

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 Seb\H{o} as well as Geelen and Kapadia: the Shortest Odd Cycle problem (SOC) in undirected graphs with conservative weights.