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Online bipartite matching with edge arrivals remained a major open question for a long time until a recent negative result by [Gamlath et al. FOCS 2019], who showed that no online policy is better than the straightforward greedy algorithm, i.e., no online algorithm has a worst-case competitive ratio better than $0.5$. In this work, we consider the bipartite matching problem with edge arrivals in a natural stochastic framework, i.e., Bayesian setting where each edge of the graph is independently realized according to a known probability distribution. We focus on a natural class of prune & greedy online policies motivated by practical considerations from a multitude of online matching platforms. Any prune & greedy algorithm consists of two stages: first, it decreases the probabilities of some edges in the stochastic instance and then runs greedy algorithm on the pruned graph. We propose prune & greedy algorithms that are $0.552$-competitive on the instances that can be pruned to a $2$-regular stochastic bipartite graph, and $0.503$-competitive on arbitrary bipartite graphs. The algorithms and our analysis significantly deviate from the prior work. We first obtain analytically manageable lower bound on the size of the matching, which leads to a non linear optimization problem. We further reduce this problem to a continuous optimization with a constant number of parameters that can be solved using standard software tools.
We provide online algorithms for secretary matching in general weighted graphs, under the well-studied models of vertex and edge arrivals. In both models, edges are associated with arbitrary weights that are unknown from the outset, and are revealed
Online bipartite matching and its variants are among the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) introduced an elegant algorithm for the unweighted problem that achieves an optimal compe
We provide prophet inequality algorithms for online weighted matching in general (non-bipartite) graphs, under two well-studied arrival models, namely edge arrival and vertex arrival. The weight of each edge is drawn independently from an a-priori kn
This article identifies a key algorithmic ingredient in the edge-weighted online matching algorithm by Zadimoghaddam (2017) and presents a simplified algorithm and its analysis to demonstrate how it works in the unweighted case.
In the stochastic online vector balancing problem, vectors $v_1,v_2,ldots,v_T$ chosen independently from an arbitrary distribution in $mathbb{R}^n$ arrive one-by-one and must be immediately given a $pm$ sign. The goal is to keep the norm of the discr