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Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities

Will Ma, Calum MacRury, Pranav Nuti

TL;DR

A new OCRS is derived that is 0.382-selectable, attaining the "independence benchmark" from the literature under the vanishing edge probabilities assumption, and positive results also apply to online matching on 1-uniform random graphs with vanishing (non-identical) edge probabilities.

Abstract

We study the performance of sequential contention resolution and matching algorithms on random graphs with vanishing edge probabilities. When the edges of the graph are processed in an adversarially-chosen order, we derive a new OCRS that is $0.382$-selectable, attaining the "independence benchmark" from the literature under the vanishing edge probabilities assumption. Complementary to this positive result, we show that no OCRS can be more than $0.390$-selectable, significantly improving upon the upper bound of $0.428$ from the literature. We also derive negative results that are specialized to bipartite graphs or subfamilies of OCRS's. Meanwhile, when the edges of the graph are processed in a uniformly random order, we show that the simple greedy contention resolution scheme which accepts all active and feasible edges is $1/2$-selectable. This result is tight due to a known upper bound. Finally, when the algorithm can choose the processing order, we show that a slight tweak to the random order -- give each vertex a random priority and process edges in lexicographic order -- results in a strictly better contention resolution scheme that is $1-\ln(2-1/e)\approx0.510$-selectable. Our positive results also apply to online matching on $1$-uniform random graphs with vanishing (non-identical) edge probabilities, extending and unifying some results from the random graphs literature.

Online Matching and Contention Resolution for Edge Arrivals with Vanishing Probabilities

TL;DR

A new OCRS is derived that is 0.382-selectable, attaining the "independence benchmark" from the literature under the vanishing edge probabilities assumption, and positive results also apply to online matching on 1-uniform random graphs with vanishing (non-identical) edge probabilities.

Abstract

We study the performance of sequential contention resolution and matching algorithms on random graphs with vanishing edge probabilities. When the edges of the graph are processed in an adversarially-chosen order, we derive a new OCRS that is -selectable, attaining the "independence benchmark" from the literature under the vanishing edge probabilities assumption. Complementary to this positive result, we show that no OCRS can be more than -selectable, significantly improving upon the upper bound of from the literature. We also derive negative results that are specialized to bipartite graphs or subfamilies of OCRS's. Meanwhile, when the edges of the graph are processed in a uniformly random order, we show that the simple greedy contention resolution scheme which accepts all active and feasible edges is -selectable. This result is tight due to a known upper bound. Finally, when the algorithm can choose the processing order, we show that a slight tweak to the random order -- give each vertex a random priority and process edges in lexicographic order -- results in a strictly better contention resolution scheme that is -selectable. Our positive results also apply to online matching on -uniform random graphs with vanishing (non-identical) edge probabilities, extending and unifying some results from the random graphs literature.
Paper Structure (14 sections, 26 theorems, 90 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 14 sections, 26 theorems, 90 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Outline of Results
  3. Techniques
  4. OCRS Positive Result
  5. OCRS for Locally-Tree-Like Edges: Proving \ref{['thm:OCRS_locally_tree_positive']}
  6. Reduction to Locally-Tree-Like Edges: Proving \ref{['thm:vanishing_to_local']}
  7. OCRS Negative Results
  8. General Graphs
  9. Trees
  10. RO-CRS and FO-CRS Positive Results
  11. Reduction to the Random Tree Case
  12. Proving Theorems \ref{['thm:ROCRS_positive']} and \ref{['thm:FOCRS_positive']}
  13. Additions to Section \ref{['sec:ocrs_positive']}
  14. Additions to Section \ref{['sec:rcrs_pos']}

Key Result

Theorem 1.1

There is a polytime OCRS that is $\frac{3 -\sqrt{5}}{2} \approx 0.382$-selectable in the vanishing regime.

Figures (2)

  • Figure 1: Graph to prove negative result. Each $W_i$ consists of $n - 2$ vertices.
  • Figure 2: Tree to prove negative result. Each $W_i$ consists of $n - 1$ vertices.

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1
  • Definition 2
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 42 more