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.

翻译：我们解决了Anari、Charikar和Ramakrishnan [ACR23] 提出的薄匹配问题，并得到了多对数因子内的结果。给定一个分数完美匹配$x$，称完美匹配$M$关于$x$是$α$-薄的，如果对任意割$(S,\overline{S})$，有$$ |M \cap E(S,\overline{S})| \leq α\cdot x(S,\overline{S}).$$ [ACR23] 猜想：对任意分数完美匹配$x$，存在一个关于$x$是$O(1)$-薄的完美匹配$M$。首先，我们证明若限制$M$在$x$的支撑集中，则$α\geq Ω(n)$，并为此设计了一个高效算法，输出一个$O(n\log n)$-薄的完美匹配，其中$n$为顶点数。随后，我们放宽该约束，证明对任意分数完美匹配$x$，存在一个完美匹配$M$（不必在$x$的支撑集中），使得$M$关于$x$是$\text{polylog}(n)$-薄的。所有结果对二部图和非二部图均成立。我们还讨论了在度量扭曲问题中的应用。