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Universal Online Contention Resolution with Preselected Order

Junyao Zhao

TL;DR

This work develops universal online contention resolution schemes (OCRSs) for matroids under arbitrary correlated priors by introducing a preselected-arrival-order model. It presents three approaches: (i) an independent-subsampling OCRS with a simple structural lemma, (ii) a correlated-subsampling OCRS that leverages permutation prefixes for stronger universality, and (iii) an LP-based optimal OCRS whose solution yields near-optimal universal guarantees and efficient computation. A key contribution is an LP-based reduction from universal OCRSs to the matroid secretary problem, enabling efficient construction from any constant-competitive secretary algorithm via ellipsoid methods and Monte Carlo coefficient estimation. The results provide computationally tractable universal OCRSs with (α,β) guarantees close to optimal, and establish a principled link between secretary algorithms and universal OCRSs across arrival models, broadening practical applications in online decision making under uncertainty.

Abstract

Online contention resolution scheme (OCRS) is a powerful technique for online decision making, which--in the case of matroids--given a matroid and a prior distribution of active elements, selects a subset of active elements that satisfies the matroid constraint in an online fashion. OCRS has been studied mostly for product distributions in the literature. Recently, universal OCRS, that works even for correlated distributions, has gained interest, because it naturally generalizes the classic notion, and its existence in the random-order arrival model turns out to be equivalent to the matroid secretary conjecture. However, currently very little is known about how to design universal OCRSs for any arrival model. In this work, we consider a natural and relatively flexible arrival model, where the OCRS is allowed to preselect (i.e., non-adaptively select) the arrival order of the elements, and within this model, we design simple and optimal universal OCRSs that are computationally efficient. In the course of deriving our OCRSs, we also discover an efficient reduction from universal online contention resolution to the matroid secretary problem for any arrival model, answering a question from Dughmi (2020).

Universal Online Contention Resolution with Preselected Order

TL;DR

This work develops universal online contention resolution schemes (OCRSs) for matroids under arbitrary correlated priors by introducing a preselected-arrival-order model. It presents three approaches: (i) an independent-subsampling OCRS with a simple structural lemma, (ii) a correlated-subsampling OCRS that leverages permutation prefixes for stronger universality, and (iii) an LP-based optimal OCRS whose solution yields near-optimal universal guarantees and efficient computation. A key contribution is an LP-based reduction from universal OCRSs to the matroid secretary problem, enabling efficient construction from any constant-competitive secretary algorithm via ellipsoid methods and Monte Carlo coefficient estimation. The results provide computationally tractable universal OCRSs with (α,β) guarantees close to optimal, and establish a principled link between secretary algorithms and universal OCRSs across arrival models, broadening practical applications in online decision making under uncertainty.

Abstract

Online contention resolution scheme (OCRS) is a powerful technique for online decision making, which--in the case of matroids--given a matroid and a prior distribution of active elements, selects a subset of active elements that satisfies the matroid constraint in an online fashion. OCRS has been studied mostly for product distributions in the literature. Recently, universal OCRS, that works even for correlated distributions, has gained interest, because it naturally generalizes the classic notion, and its existence in the random-order arrival model turns out to be equivalent to the matroid secretary conjecture. However, currently very little is known about how to design universal OCRSs for any arrival model. In this work, we consider a natural and relatively flexible arrival model, where the OCRS is allowed to preselect (i.e., non-adaptively select) the arrival order of the elements, and within this model, we design simple and optimal universal OCRSs that are computationally efficient. In the course of deriving our OCRSs, we also discover an efficient reduction from universal online contention resolution to the matroid secretary problem for any arrival model, answering a question from Dughmi (2020).

Paper Structure

This paper contains 27 sections, 28 theorems, 81 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Overview of our universal OCRSs
  3. Universal OCRSs from secretary algorithms
  4. Preliminaries
  5. Matroids
  6. Contention resolution schemes
  7. Other useful notions and lemmata
  8. A simple universal OCRS via independent subsampling
  9. Main structural lemma
  10. Constructing our universal OCRS with independent subsampling
  11. An improved universal OCRS via correlated subsampling
  12. Main structural lemma
  13. Correlated subsampling
  14. Constructing our universal OCRS with correlated subsampling
  15. An optimal universal OCRS via linear programming
  16. ...and 12 more sections

Key Result

Theorem 1.1

For all $\alpha\in[0,1]$, Algorithm alg:universal_ocrs_independent_subsampling is an $(\alpha,\frac{\alpha^2}{4})$-universal OCRS, and Algorithm alg:universal_ocrs_correlated_subsampling is an $(\alpha,\frac{\alpha^2}{2})$-universal OCRS. Both algorithms preselect the arrival order of the elements b

Figures (1)

  • Figure 1: Illustration of Example \ref{['ex:independent_subsampling_lower_bound']}.

Theorems & Definitions (61)

  • Theorem 1.1: Restatement of Theorem \ref{['thm:universal_ocrs_independent_subsampling']} and Theorem \ref{['thm:universal_ocrs_correlated_subsampling']}
  • Theorem 1.2: Informal restatement of Theorem \ref{['thm:universal_ocrs_linear_programming']}
  • Theorem 1.3: Informal restatement of Theorem \ref{['thm:secretary_to_ocrs']}
  • Definition 2.1: matroid
  • Definition 2.2: rank
  • Definition 2.3: span
  • Definition 2.4: basis
  • Definition 2.5: restriction
  • Lemma 2.6
  • Example 2.7
  • ...and 51 more