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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 known probability distribution. Under edge arrival, the weight of each edge is revealed upon arrival, and the algorithm decides whether to include it in the matching or not. Under vertex arrival, the weights of all edges from the newly arriving vertex to all previously arrived vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. To study these settings, we introduce a novel unified framework of batched prophet inequalities that captures online settings where elements arrive in batches; in particular it captures matching under the two aforementioned arrival models. Our algorithms rely on the construction of suitable online contention resolution scheme (OCRS). We first extend the framework of OCRS to batched-OCRS, we then establish a reduction from batched prophet inequality to batched OCRS, and finally we construct batched OCRSs with selectable ratios of 0.337 and 0.5 for edge and vertex arrival models, respectively. Both results improve the state of the art for the corresponding settings. For the vertex arrival, our result is tight. Interestingly, a pricing-based prophet inequality with comparable competitive ratios is unknown.
The rich literature on online Bayesian selection problems has long focused on so-called prophet inequalities, which compare the gain of an online algorithm to that of a prophet who knows the future. An equally-natural, though significantly less well-
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 o
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
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.
The setting of the classic prophet inequality is as follows: a gambler is shown the probability distributions of $n$ independent, non-negative random variables with finite expectations. In their indexed order, a value is drawn from each distribution,