Random-order Contention Resolution via Continuous Induction: Tightness for Bipartite Matching under Vertex Arrivals

We introduce a new approach for designing Random-order Contention Resolution Schemes (RCRS) via exact solution in continuous time. Given a function $c(y):[0,1] \rightarrow [0,1]$, we show how to select each element which arrives at time $y \in [0,1]$ with probability exactly $c(y)$. We provide a rigorous algorithmic framework for achieving this, which discretizes the time interval and also needs to sample its past execution to ensure these exact selection probabilities. We showcase our framework in the context of online contention resolution schemes for matching with random-order vertex arrivals. For bipartite graphs with two-sided arrivals, we design a $(1+e^{-2})/2 \approx 0.567$-selectable RCRS, which we also show to be tight. Next, we show that the presence of short odd-length cycles is the only barrier to attaining a (tight) $(1+e^{-2})/2$-selectable RCRS on general graphs. By generalizing our bipartite RCRS, we design an RCRS for graphs with odd-length girth $g$ which is $(1+e^{-2})/2$-selectable as $g \rightarrow \infty$. This convergence happens very rapidly: for triangle-free graphs (i.e., $g \ge 5$), we attain a $121/240 + 7/16 e^2 \approx 0.563$-selectable RCRS. Finally, for general graphs we improve on the $8/15 \approx 0.533$-selectable RCRS of Fu et al. (ICALP, 2021) and design an RCRS which is at least $0.535$-selectable. Due to the reduction of Ezra et al. (EC, 2020), our bounds yield a $0.535$-competitive (respectively, $(1+e^{-2})/2$-competitive) algorithm for prophet secretary matching on general (respectively, bipartite) graphs under vertex arrivals

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

Random order online contention resolution schemes (ROCRS) are structured online rounding algorithms with numerous applications and links to other well-known online selection problems, like the matroid secretary conjecture. We are interested in ROCRS subject to a matroid constraint, which is among the most studied constraint families. Previous ROCRS required to know upfront the full fractional point to be rounded as well as the matroid. It is unclear to what extent this is necessary. Fu, Lu, Tang, Turkieltaub, Wu, Wu, and Zhang (SOSA 2022) shed some light on this question by proving that no strong (constant-selectable) online or even offline contention resolution scheme exists if the fractional point is unknown, not even for graphic matroids. In contrast, we show, in a setting with slightly more knowledge and where the fractional point reveals one by one, that there is hope to obtain strong ROCRS by providing a simple constant-selectable ROCRS for graphic matroids that only requires to know the size of the ground set in advance. Moreover, our procedure holds in the more general adversarial order with a sample setting, where, after sampling a random constant fraction of the elements, all remaining (non-sampled) elements may come in adversarial order.Comment: To be published in SOSA2

Algorithms to Approximate Column-Sparse Packing Problems

Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go "half the remaining distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM Transactions of Algorithm