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Towards an Optimal Contention Resolution Scheme for Matchings

Pranav Nuti, Jan Vondrák

TL;DR

The paper advances contention resolution schemes for matchings by delivering a near-optimal $c$-balanced CRS for general matchings in the vanishing-probability regime, and a strong $0.509$-balanced CRS for bipartite matchings, establishing separations between offline vs online and monotone vs non-monotone strategies. The core technique combines a refined KS analysis on random trees, a coupling between $R(x)$ and Galton–Watson trees, and, for the bipartite case, a multi-stage red/blue/gray scheme augmented by the FKG inequality to handle correlations. These results sharpen our understanding of the correlation gap for bipartite and general matchings and connect to van der Waerden’s conjecture through a probabilistic lens. Practically, the schemes enable robust rounding of fractional matchings with improved guarantees, and they extend to a combinatorial allocation problem via a Configuration LP and demand oracles.

Abstract

In this paper, we study contention resolution schemes for matchings. Given a fractional matching $x$ and a random set $R(x)$ where each edge $e$ appears independently with probability $x_e$, we want to select a matching $M \subseteq R(x)$ such that $\Pr[e \in M \mid e \in R(x)] \geq c$, for $c$ as large as possible. We call such a selection method a $c$-balanced contention resolution scheme. Our main results are (i) an asymptotically (in the limit as $\|x\|_\infty$ goes to 0) optimal $\simeq 0.544$-balanced contention resolution scheme for general matchings, and (ii) a $0.509$-balanced contention resolution scheme for bipartite matchings. To the best of our knowledge, this result establishes for the first time, in any natural relaxation of a combinatorial optimization problem, a separation between (i) offline and random order online contention resolution schemes, and (ii) monotone and non-monotone contention resolution schemes. We also present an application of our scheme to a combinatorial allocation problem, and discuss some open questions related to van der Waerden's conjecture for the permanent of doubly stochastic matrices.

Towards an Optimal Contention Resolution Scheme for Matchings

TL;DR

The paper advances contention resolution schemes for matchings by delivering a near-optimal -balanced CRS for general matchings in the vanishing-probability regime, and a strong -balanced CRS for bipartite matchings, establishing separations between offline vs online and monotone vs non-monotone strategies. The core technique combines a refined KS analysis on random trees, a coupling between and Galton–Watson trees, and, for the bipartite case, a multi-stage red/blue/gray scheme augmented by the FKG inequality to handle correlations. These results sharpen our understanding of the correlation gap for bipartite and general matchings and connect to van der Waerden’s conjecture through a probabilistic lens. Practically, the schemes enable robust rounding of fractional matchings with improved guarantees, and they extend to a combinatorial allocation problem via a Configuration LP and demand oracles.

Abstract

In this paper, we study contention resolution schemes for matchings. Given a fractional matching and a random set where each edge appears independently with probability , we want to select a matching such that , for as large as possible. We call such a selection method a -balanced contention resolution scheme. Our main results are (i) an asymptotically (in the limit as goes to 0) optimal -balanced contention resolution scheme for general matchings, and (ii) a -balanced contention resolution scheme for bipartite matchings. To the best of our knowledge, this result establishes for the first time, in any natural relaxation of a combinatorial optimization problem, a separation between (i) offline and random order online contention resolution schemes, and (ii) monotone and non-monotone contention resolution schemes. We also present an application of our scheme to a combinatorial allocation problem, and discuss some open questions related to van der Waerden's conjecture for the permanent of doubly stochastic matrices.
Paper Structure (25 sections, 17 theorems, 68 equations, 1 figure)

This paper contains 25 sections, 17 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our results
  3. Contention resolution for matchings.
  4. Our techniques
  5. Parallels with van der Waerden's conjecture
  6. van der Waerden's conjecture
  7. van der Waerden's conjecture, reformulated
  8. An optimal contention resolution scheme for vanishing probabilities
  9. The Karp-Sipser Algorithm
  10. Random Trees
  11. The Karp-Sipser algorithm on trees
  12. L/W-labeling algorithm.
  13. The Karp-Sipser algorithm on random trees
  14. Putting it all together
  15. Improved contention resolution schemes for bipartite matchings
  16. ...and 10 more sections

Key Result

Theorem 1

Suppose that $x$ is a fractional matching such that $\|x\|_\infty \leq \epsilon$. Then, there is a function $f$ (not dependent on $x$) with $\lim_{\epsilon \rightarrow 0} f(\epsilon) = 0$, and a $(\gamma - f(\epsilon))$-balanced contention resolution scheme, where $\gamma \simeq 0.544$ is the imposs

Figures (1)

  • Figure 1: Bipartite graph to establish Theorem \ref{['theorem:CRS-1']} with $|E| = 2$.

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Conjecture 1
  • Conjecture 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 24 more