Online Matching with Stochastic Rewards

Mehta, Aranyak; Panigrahi, Debmalya

doi:10.1109/focs.2012.65

Cited by 62 publications

(88 citation statements)

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“…In [24], the authors first show that a Greedy algorithm (which matches the arriving vertex j to that available neighbor i which has the highest value of p ij ) achieves a ratio of 1/2. Thus, 1/2 is a baseline for this problem upon which to improve (this is the case in the classic version as well, although the argument for the stochastic setting is a strict generalization).…”

Section: Previous Resultsmentioning

confidence: 99%

“…Equal probabilities case: [24] focuses on the special case in which all the probabilities are equal, i.e., the p ij are either 0 or p, for some value p ∈ [0, 1]. They provide an algorithm, called StochasticBalance to solve this special case, giving a competitive ratio of 0.5(1 + e −2 ) 0.567 as p → 0, and which reduces to 1/2 as p increases to 1.…”

Section: Previous Resultsmentioning

confidence: 99%

“…They provide an algorithm, called StochasticBalance to solve this special case, giving a competitive ratio of 0.5(1 + e −2 ) 0.567 as p → 0, and which reduces to 1/2 as p increases to 1. This algorithm works as follows: Algorithm 1.1: StochasticBalance (For the equal probabilities case, from [24]. )…”

Section: Previous Resultsmentioning

confidence: 99%

“…A recent paper [24] introduced a new variant of this classic problem, which they called the OnlineStochasticMatching problem. The motivation behind this variant was the observation that, in both the applications cited above, the goal is not simply to maximize the size of the matching, but to maximize the number of matched edges that become successful.…”

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confidence: 99%

“…In [24], the authors provide a definition for the competitive ratio of this problem, which captures the performance of the algorithm with respect to an optimum benchmark. Because the goal is to study both the online and the stochastic aspects of the problem, they define OPT as the optimal solution to the following offline, deterministic problem: The graph is known in advance and the reward of an edge (i, j) is deterministically p ij .…”

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Online Stochastic Matching with Unequal Probabilities

Mehta¹,

Waggoner²,

Zadimoghaddam³

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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The online stochastic matching problem is a variant of online bipartite matching in which edges are labeled with probabilities. A match will "succeed" with the probability along that edge; this models, for instance, the click of a user in search advertisement. The goal is to maximize the expected number of successful matches. This problem was introduced by Mehta and Panigrahi (FOCS 2012), who focused on the case where all probabilities in the graph are equal. They gave a 0.567-competitive algorithm for vanishing probabilities, relative to a natural benchmark, leaving the general case as an open question.This paper examines the general case where the probabilities may be unequal. We take a new algorithmic approach rather than generalizing that of Mehta and Panigrahi: Our algorithm maintains, at each time, the probability that each offline vertex has succeeded thus far, and chooses assignments so as to maximize marginal contributions to these probabilities. When the algorithm does not observe the realizations of the edges, this approach gives a 0.5-competitive algorithm, which achieves the known upper bound for such "non-adaptive" algorithms. We then modify this approach to be "semi-adaptive:" if the chosen target has already succeeded, choose the arrival's "second choice" instead (while still updating the probabilities non-adaptively). With one additional tweak to control the analysis, we show that this algorithm achieves a competitive ratio of 0.534 for the unequal, vanishing probabilities setting. A "fully-adaptive" version of this algorithm turns out to be identical to an algorithm proposed, but not analyzed, in Mehta and Panigrahi (2012); we do not manage to analyze it either since it introduces too many dependencies between the stochastic processes. Our semi-adaptive algorithm thus can be seen as allowing analysis of competitive ratio while still capturing the power of adaptivity.

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Section: Previous Resultsmentioning

confidence: 99%

Section: Previous Resultsmentioning

confidence: 99%

Section: Previous Resultsmentioning

confidence: 99%

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confidence: 99%

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