Core Concepts
This paper analyzes online contention resolution schemes (OCRS) and random-order contention resolution schemes (RCRS) for resource constraints defined by matchings in graphs. The authors improve the state-of-the-art algorithmic guarantees and derive new impossibility results for these schemes.
Abstract
The paper focuses on online contention resolution schemes (OCRS) and random-order contention resolution schemes (RCRS) for resource constraints defined by matchings in graphs.
Key highlights:
For OCRS, the authors analyze and improve the algorithm of Ezra et al. (2022):
For general graphs, they show the algorithm is 0.344-selectable, improving the previous 0.337 guarantee.
For bipartite graphs, they show the algorithm is 0.349-selectable.
They also derive new impossibility results, showing no OCRS can be more than 0.4-selectable, and the algorithm of Ezra et al. (2022) is no more than 0.361-selectable.
For RCRS, the authors provide new algorithms with improved selectability guarantees:
For general graphs, they present a 0.474-selectable RCRS.
For bipartite graphs without 3-cycles or 5-cycles, they present a 0.478-selectable RCRS.
They also show a fundamental barrier that no RCRS can be more than 1/2-selectable.
The authors use a variety of techniques, including the FKG inequality, analytical optimization, and reductions to 1-regular inputs, to derive these results. Their analyses reveal interesting structural properties of the worst-case configurations for these schemes.
Overall, the paper significantly advances the state-of-the-art for online contention resolution on matching polytopes, with implications for various online resource allocation problems.
Stats
P[blocked(e)] ≥ c
P[matchedu(e) ∩ matchedv(e)] ≤ b2 (k∑i=1 yi - byi + by2i/(1 + byi)) (k∑i=1 zi - bzi + bz2i/(1 + bzi)) - b2 (k∑i=1 yi - byi + by2i/(1 + byi))(k∑i=1 zi - bzi + bz2i/(1 + bzi))/(1 + byi')(1 + bzi')
P[|Re| = 0 | Ye = y] = ∏f∈∂(e) ℓ(xf,y)
P[sblf(h) | Re = {f}, Ye = y] ≥ s(xh)/(2(1 - xh) - xf - xfc)(1 - (1 - e^(-(2(1 - xh) - xf - xfc)y))/(2(1 - xh) - xf - xfc)y)
Quotes
"Online Contention Resolution Schemes (OCRS's) represent a modern tool for selecting a subset of elements, subject to resource constraints, when the elements are presented to the algorithm sequentially."
"We improve the state of the art for all combinations of variants, both in terms of algorithmic guarantees and impossibility results."
"Our results for OCRS directly improve the best-known competitive ratios for online accept/reject, probing, and pricing problems on graphs in a unified manner."