An Optimal Online Algorithm for Weighted Bipartite Matching and Extensions to Combinatorial Auctions

Keßelheim, Thomas; Radke, Klaus; Tönnis, Andreas; Vöcking, Berthold

doi:10.1007/978-3-642-40450-4_50

Cited by 105 publications

(148 citation statements)

References 20 publications

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“…Other variants in the non-stochastic version include vertex-weighted matching [1] and edge-weighted matching ( [9], among others). Another series of results, also under the name of Stochastic Matching, deals with stochastic arrivals, either random order, or iid ( [8,10,2,16,21,22,20,19]) for the different versions of matching or budgeted allocation. Here, the rewards are non-stochastic, and the problems admit better ratios than the adversarial arrival models (as opposed to our problem which is provably strictly harder than the non-stochastic version).…”

Section: Previous Resultsmentioning

confidence: 99%

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%

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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“…(2) Under what conditions can a secretary algorithm be modified to solve the edge-weighted online bipartite matching problem with random arrival order [15,12]?…”

Section: Finitementioning

confidence: 99%

“…Kesselheim et al [12] essentially showed that the (1, 1)-threshold algorithm can be applied to the edgeweighted online 1-matching problem to get a performance ratio of 1 e . Independent of our work, in a recent paper Kesselheim et al [13] considered a general class of online packing LPs.…”

Section: Finitementioning

confidence: 99%

“…We use a similar algorithm as in [12], in which at every step, a K-matching is formed between all arrived online nodes and offline nodes. However, we need a sophisticated objective function to form this K-matching in order to relate the optimal K-matching in hindsight with the expected payoff of (K, K)-secretary problem.…”

Section: Finitementioning

confidence: 99%

“…Conditioning on L i and v i . As argued in [12], we condition on the set L i and node v i and show that the remaining randomness of the relative order among nodes in L i−1 is enough to give the desired lower bound on the probability.…”

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confidence: 99%

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Revealing Optimal Thresholds for Generalized Secretary Problem via Continuous LP: Impacts on Online K-Item Auction and Bipartite K-Matching with Random Arrival Order

Chan¹,

Chen²,

Jiang³

2014

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

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We consider the general (J, K)-secretary problem, where n totally ordered items arrive in a random order. An algorithm observes the relative merits of arriving items and is allowed to make J selections. The objective is to maximize the expected number of items selected among the K best items.Buchbinder, Jain and Singh proposed a finite linear program (LP) that completely characterizes the problem, but it is difficult to analyze the asymptotic behavior of its optimal solution as n tends to infinity. Instead, we prove a formal connection between the finite model and an infinite model, where there are a countably infinite number of items, each of which has arrival time drawn independently and uniformly from [0, 1].The finite LP extends to a continuous LP, whose complementary slackness conditions reveal an optimal algorithm which involves JK thresholds that play a similar role as the 1 e -threshold in the optimal classical secretary algorithm. In particular, for the case K = 1, the J optimal thresholds have a nice "rational description". Our continuous LP analysis gives a very clear perspective on the problem, and the new insights inspire us to solve two related problems.1. We settle the open problem whether algorithms based only on relative merits can achieve optimal ratio for matroid secretary problems. We show that, for online 2-item auction with random arriving bids (the K-uniform matroid problem with K = 2), an algorithm making decisions based only on relative merits cannot achieve the optimal ratio. This is in contrast with the folklore that, for online 1-item auction, no algorithm can have performance ratio strictly larger than 1 e , which is achievable by an algorithm that considers only relative merits. 2. We give a general transformation technique that takes any monotone algorithm (such as threshold

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Packing returning secretaries

Hoefer

Wilhelmi

2020

Networks

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We study online secretary problems with returns in combinatorial packing domains with n candidates that arrive sequentially over time in random order. The goal is to determine a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n arrivals occur in random order. We propose a simple 0.5‐competitive algorithm. For the online bipartite matching problem, we obtain an algorithm with ratio at least 0.5721 − o(1), and an algorithm with ratio at least 0.5459 for all n ≥ 1. We extend all algorithms and ratios to k ≥ 2 arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed. We focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Θ(n log n) is always sufficient. For bipartite matching, we can show a tight bound of O(r log n), where r is the size of the optimum matching. For matroids, we can improve this further to a tight bound of O(r′ log(n/r′)), where r′ is the minimum rank of the matroid and the dual matroid.

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An Optimal Online Algorithm for Weighted Bipartite Matching and Extensions to Combinatorial Auctions

Cited by 105 publications

References 20 publications

Online Stochastic Matching with Unequal Probabilities

Online Stochastic Matching with Unequal Probabilities

Revealing Optimal Thresholds for Generalized Secretary Problem via Continuous LP: Impacts on Online K-Item Auction and Bipartite K-Matching with Random Arrival Order

Packing returning secretaries

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