Towards an Optimal Contention Resolution Scheme for Matchings

11/07/2022
by   Pranav Nuti, et al.
0

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 ⊆ R(x) such that [e ∈ M | e ∈ R(x)] ≥ 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_∞ goes to 0) optimal ≃ 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/25/2021

An Optimal Monotone Contention Resolution Scheme for Uniform and Partition Matroids

A common approach to solve a combinatorial optimization problem is to fi...
research
05/21/2019

An Optimal Monotone Contention Resolution Scheme for Bipartite Matchings via a Polyhedral Viewpoint

Relaxation and rounding approaches became a standard and extremely versa...
research
06/25/2018

Optimal Online Contention Resolution Schemes via Ex-Ante Prophet Inequalities

Online contention resolution schemes (OCRSs) were proposed by Feldman, S...
research
03/06/2021

Matroid Secretary is Equivalent to Contention Resolution

We show that the matroid secretary problem is equivalent to correlated c...
research
11/20/2021

Oblivious Online Contention Resolution Schemes

Contention resolution schemes (CRSs) are powerful tools for obtaining "e...
research
01/08/2023

Fully Dynamic Online Selection through Online Contention Resolution Schemes

We study fully dynamic online selection problems in an adversarial/stoch...
research
11/28/2022

Simple Random Order Contention Resolution for Graphic Matroids with Almost no Prior Information

Random order online contention resolution schemes (ROCRS) are structured...

Please sign up or login with your details

Forgot password? Click here to reset