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Register a new userOnline Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals
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Xiaowei Wu
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival.
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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 competitive ratio of $1-1/e$. Later, Aggarwal et al. (SODA 2011) generalized their algorithm and analysis to the vertex-weighted case. Little is known, however, about the most general edge-weighted problem aside from the trivial $1/2$-competitive greedy algorithm. In this paper, we present the first online algorithm that breaks the long-standing $1/2$ barrier and achieves a competitive ratio of at least $0.5086$. In light of the hardness result of Kapralov, Post, and Vondrak (SODA 2013) that restricts beating a $1/2$ competitive ratio for the more general problem of monotone submodular welfare maximization, our result can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in the online setting. The main ingredient in our online matching algorithm is a novel subroutine called online correlated selection (OCS), which takes a sequence of pairs of vertices as input and selects one vertex from each pair. Instead of using a fresh random bit to choose a vertex from each pair, the OCS negatively correlates decisions across different pairs and provides a quantitative measure on the level of correlation. We believe our OCS technique is of independent interest and will find further applications in other online optimization problems.
We study the oblivious matching problem, which aims at finding a maximum matching on a graph with unknown edge set. Any algorithm for the problem specifies an ordering of the vertex pairs. The matching is then produced by probing the pairs following the ordering, and including a pair if both of them are unmatched and there exists an edge between them. The unweighted (Chan et al. (SICOMP 2018)) and the vertex-weighted (Chan et al. (TALG 2018
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.
Let $G=(V, E)$ be a given edge-weighted graph and let its {em realization} $mathcal{G}$ be a random subgraph of $G$ that includes each edge $e in E$ independently with probability $p$. In the {em stochastic matching} problem, the goal is to pick a sparse subgraph $Q$ of $G$ without knowing the realization $mathcal{G}$, such that the maximum weight matching among the realized edges of $Q$ (i.e. graph $Q cap mathcal{G}$) in expectation approximates the maximum weight matching of the whole realization $mathcal{G}$. In this paper, we prove that for any desirably small $epsilon in (0, 1)$, every graph $G$ has a subgraph $Q$ that guarantees a $(1-epsilon)$-approximation and has maximum degree only $O_{epsilon, p}(1)$. That is, the maximum degree of $Q$ depends only on $epsilon$ and $p$ (both of which are known to be necessary) and not for example on the number of nodes in $G$, the edge-weights, etc. The stochastic matching problem has been studied extensively on both weighted and unweighted graphs. Previously, only existence of (close to) half-approximate subgraphs was known for weighted graphs [Yamaguchi and Maehara, SODA18; Behnezhad et al., SODA19]. Our result substantially improves over these works, matches the state-of-the-art for unweighted graphs [Behnezhad et al., STOC20], and essentially settles the approximation factor.
We study the greedy-based online algorithm for edge-weighted matching with (one-sided) vertex arrivals in bipartite graphs, and edge arrivals in general graphs. This algorithm was first studied more than a decade ago by Korula and Pal for the bipartite case in the random-order model. While the weighted bipartite matching problem is solved in the random-order model, this is not the case in recent and exciting online models in which the online player is provided with a sample, and the arrival order is adversarial. The greedy-based algorithm is arguably the most natural and practical algorithm to be applied in these models. Despite its simplicity and appeal, and despite being studied in multiple works, the greedy-based algorithm was not fully understood in any of the studied online models, and its actual performance remained an open question for more than a decade. We provide a thorough analysis of the greedy-based algorithm in several online models. For vertex arrivals in bipartite graphs, we characterize the exact competitive-ratio of this algorithm in the random-order model, for any arrival order of the vertices subsequent to the sampling phase (adversarial and random orders in particular). We use it to derive tight analysis in the recent adversarial-order model with a sample (AOS model) for any sample size, providing the first result in this model beyond the simple secretary problem. Then, we generalize and strengthen the black box method of converting results in the random-order model to single-sample prophet inequalities, and use it to derive the state-of-the-art single-sample prophet inequality for the problem. Finally, we use our new techniques to analyze the greedy-based algorithm for edge arrivals in general graphs and derive results in all the mentioned online models. In this case as well, we improve upon the state-of-the-art single-sample prophet inequality.
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