On Thin Perfect Matchings up to Polylogarithmic Factors

2026-05-31

AI Summary

This work addresses the “thin perfect matching” problem posed by Anari, Charikar, and Ramakrishnan—namely, whether for any fractional perfect matching \( x \), there exists a perfect matching that is \( O(1) \)-thin with respect to \( x \). The paper presents a unified treatment of both bipartite and non-bipartite graphs, distinguishing between matchings supported inside and outside the support of \( x \). Leveraging tools from combinatorial optimization, cut analysis, and fractional matching theory, it achieves a breakthrough by reducing the thinness factor in the support-outside case to \( \mathrm{polylog}(n) \), improving upon the previous \( \Omega(n) \) lower bound. Additionally, the authors design an algorithm yielding an \( O(n \log n) \)-thin matching within the support and establish the existence of \( \mathrm{polylog}(n) \)-thin perfect matchings, further exploring their implications for metric distortion problems.

Abstract

We resolve the thin matching problem proposed by Anari, Charikar and Ramakrishnan [ACR23] up to polylogarithmic factors. Given a fractional perfect matching $x$, we say a perfect matching $M$ is $α$-thin w.r.t. $x$ if for any cut $(S,\overline{S})$, we have $$ |M \cap E(S,\overline{S})| \leq α\cdot x(S,\overline{S}).$$ [ACR23] conjectured that for any fractional perfect matching $x$, there exists a perfect matching $M$ which is $O(1)$-thin w.r.t. $x$. First, we show that if $M$ is restricted to be in the support of $x$, then $α\geq Ω(n)$ and we complement this by designing an efficient algorithm that outputs an $O(n\log n)$-thin perfect matching where $n$ is the number of vertices. Then, we relax this constraint and show that for any fractional perfect matching $x$, there is a perfect matching $M$ (which is not necessarily in the support of $x$) such that $M$ is $\text{polylog}(n)$-thin w.r.t. $x$. All results work for both bipartite and non-bipartite graphs. We also discuss applications to the metric distortion problem.

Problem

• thin matching
• fractional perfect matching
• graph cuts
• polylogarithmic factors
• metric distortion

Innovation

• thin matching
• fractional perfect matching
• polylogarithmic approximation
• graph cuts
• metric distortion