On Thin Perfect Matchings up to Polylogarithmic Factors

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.

• The paper proves a sharp dichotomy: support-constrained perfect matchings require Ω(n) thinness but admit randomized O(n log n)-thin solutions, while unrestricted matchings achieve polylogarithmic thinness.
• The paper combines trimming, randomized edge sampling, Chernoff bounds, cut counting, and Tutte-witness certification for support-constrained matchings, while using tree-cut sparsifiers and discrepancy-optimal greedy pairing in the relaxed setting.
• The paper applies the relaxed theorem to derandomize ordinal metric matching mechanisms, reducing the previous O(n²) distortion overhead to O(log^{5/2} n log log n) times the randomized mechanism’s distortion.

Overview

The paper studies the thin perfect matching problem introduced by Anari, Charikar, and Ramakrishnan (2606.01330). Given a weighted graph $G=(V,E,x)$ where $x$ is a fractional perfect matching (a point in the perfect matching polytope $\mathcal{PM}(G)$), a perfect matching $M$ is $\alpha$-thin with respect to $x$ if for every nonempty $S \subsetneq V$,

$|M \cap \delta(S)| \leq \alpha \cdot x(\delta(S)).$

The motivating conjecture of Anari et al. posits that every fractional perfect matching admits an $O(1)$-thin perfect matching. This paper resolves the problem up to polylogarithmic factors and establishes a sharp structural dichotomy depending on whether the matching is required to lie inside the support of $x$. All results hold for both bipartite and general graphs.

Linear lower bound within the support

The first contribution is a worst-case separation between the thin matching and thin spanning tree settings. The authors construct a bipartite graph on $x$0 vertices with a fractional perfect matching $x$1 in which solid edges carry weight $x$2 and dotted edges carry weight $x$3, such that any perfect matching must use at least one dotted edge. Each dotted edge crosses a cut of value $x$4, so any set containing it has thinness at least $x$5. Consequently:

Lower bound: there exists $x$6 such that every perfect matching $x$7 satisfies $x$8.

This contrasts sharply with the thin tree conjecture, where one expects an $x$9-thin tree contained in the support of any point in the spanning tree polytope. It implies that support-constrained thin matchings are intrinsically harder: no constant thinness is achievable within the support, so the natural goal becomes near-linear thinness.

Support-constrained $\mathcal{PM}(G)$0-thin matching

Complementing the lower bound, the paper gives a randomized polynomial-time algorithm that outputs a perfect matching $\mathcal{PM}(G)$1 that is $\mathcal{PM}(G)$2-thin — only an $\mathcal{PM}(G)$3 loss over the worst-case optimum. The algorithm proceeds in two stages:

Trimming: edges are iteratively deleted from cuts $\mathcal{PM}(G)$4 of connected components whose fractional value is at most $\mathcal{PM}(G)$5. Since each trimming step removes weight at most $\mathcal{PM}(G)$6 and increases the component count by one, the total deleted weight is at most $\mathcal{PM}(G)$7. Every resulting component is then $\mathcal{PM}(G)$8-edge connected under the trimmed weights.

Sampling and Tutte certification: each edge $\mathcal{PM}(G)$9 is independently sampled $M$0 times with probability $M$1. Thinness of the sampled graph follows from a Chernoff bound combined with a Karger-style cut-counting argument. Existence of a perfect matching is certified via Tutte's theorem: the authors bound the number of Tutte witnesses by $M$2, show that any witness requires a set $M$3 of crossing edges with $M$4 (using odd-set constraints of the perfect matching polytope together with the trimming budget), and conclude that all witnesses are simultaneously destroyed with probability at least $M$5.

A notable remark is that the sampling parameter can be improved to $M$6 while preserving the guarantee, tightening the gap to the $M$7 lower bound to a single logarithmic factor.

Polylogarithmic relaxed thin matching via tree-cut sparsifiers

When the matching need not respect the support of $M$8, the picture changes qualitatively. The main theorem states that for any fractional perfect matching $M$9 on a graph with $\alpha$0 vertices (bipartite or not), there exists a perfect matching $\alpha$1 (over the complete feasibility graph) that is $\alpha$2-thin with respect to $\alpha$3.

The proof architecture combines two ingredients:

Tree-cut sparsification: the authors invoke recent single-tree cut sparsifier results to obtain a weighted tree $\alpha$4 (with Steiner nodes) such that for every cut $\alpha$5, the minimum tree cut projecting onto it satisfies $\alpha$6 with $\alpha$7. The tree thus captures all cut structure up to a polylogarithmic factor while discarding edge-level detail.

Greedy tree pairing: since only tree cuts matter, it suffices to control the load of $\alpha$8 on each tree edge $\alpha$9, i.e., the number of matched pairs whose tree path uses $x$0. A bottom-up greedy pairing algorithm achieves the exact identity $x$1 for every tree edge, where $x$2 is the discrepancy of $x$3: the imbalance $x$4 in the bipartite case, or the parity indicator in the general case. A charging argument shows that any matching must have load at least this discrepancy across each tree cut, and Lemma (discrepancy lower bound) shows $x$5 using degree constraints and odd-set constraints respectively. Combining these yields $x$6 against tree cuts, hence $x$7-thinness overall through the sparsifier quality $x$8.

An important caveat noted by the authors: the greedy pairing matches vertices purely according to their positions in the tree sparsifier and does not respect the support of $x$9; the construction relies essentially on the freedom to use arbitrary edges of the complete (or complete bipartite) feasibility graph.

Application to derandomizing metric distortion mechanisms

In the metric distortion problem from social choice theory, a mechanism receives only ordinal preferences of $S \subsetneq V$0 buyers over $S \subsetneq V$1 items with metric costs $S \subsetneq V$2 and outputs a perfect matching; its distortion is the worst-case ratio of its expected cost to the optimal matching cost. Anari et al. showed that if every fractional matching admits a $S \subsetneq V$3-thin perfect matching, then any randomized mechanism with expected distortion $S \subsetneq V$4 can be converted into a deterministic mechanism with distortion $S \subsetneq V$5.

Combining the relaxed thin matching theorem with that reduction yields a deterministic mechanism with distortion $S \subsetneq V$6. Thus, any randomized ordinal matching mechanism can be derandomized with only polylogarithmic loss in distortion — resolving, up to polylogarithmic factors, the derandomization question raised by Anari et al., which previously incurred an $S \subsetneq V$7 overhead.

Limitations and open questions

The authors state explicitly that neither thinness bound is expected to be tight. Two concrete questions remain open: whether the $S \subsetneq V$8 factor in the support-constrained setting can be reduced toward the $S \subsetneq V$9 lower bound (i.e., eliminating the logarithmic gap), and whether the $|M \cap \delta(S)| \leq \alpha \cdot x(\delta(S)).$0 dependence in the relaxed setting can be removed entirely, which would settle the original constant-thinness conjecture of Anari, Charikar, and Ramakrishnan. Additionally, the derandomization corollary inherits both the sparsifier quality $|M \cap \delta(S)| \leq \alpha \cdot x(\delta(S)).$1 and the reduction's extra $|M \cap \delta(S)| \leq \alpha \cdot x(\delta(S)).$2 factor, so improving either component would directly improve the deterministic distortion guarantee.

Conclusion

This paper settles the thin matching problem of Anari, Charikar, and Ramakrishnan up to polylogarithmic factors, with a clean dichotomy: within the support of $|M \cap \delta(S)| \leq \alpha \cdot x(\delta(S)).$3, thinness is necessarily $|M \cap \delta(S)| \leq \alpha \cdot x(\delta(S)).$4 and achievable at $|M \cap \delta(S)| \leq \alpha \cdot x(\delta(S)).$5; without the support constraint, $|M \cap \delta(S)| \leq \alpha \cdot x(\delta(S)).$6-thin matchings always exist. Technically, the work pairs a Tutte-witness counting argument for the support-constrained regime with a novel reduction from thin matchings to single-tree cut sparsifiers plus a discrepancy-optimal greedy pairing in the relaxed regime. As a direct consequence, randomized metric distortion mechanisms admit deterministic counterparts with polylogarithmic distortion overhead.