Robust Matchings and Matroid Intersections

Fujita, Ryo; Kobayashi, Yusuke; Makino, Kazuhisa

doi:10.1137/100808800

Cited by 11 publications

(11 citation statements)

References 4 publications

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“…The idea of finding a combinatorial object (a weighted matching in this case) that is robust against an adversarial choice of cardinality has also been studied on other domains. Fujita, Kobayashi, and Makino [5] proved that the above results for matchings hold for the problem of finding common independent sets of two matroids. They also showed that computing the maximum robustness factor α for a given instance is NP-hard even for the case of matchings in bipartite graphs.…”

Section: Related Workmentioning

confidence: 85%

“…The proof involves a sequence of technical arguments for showing that the worst case is attained for graphs with 3 or 5 edges. Fujita, Kobayashi, and Makino's [5] generalization of this result also makes use of an optimization problem over the weights, additionally using the dual of the matroid intersection polytope to split the weight of each edge in two separate parts. We give a new proof of Hassin and Rubinstein's original result based on LP duality.…”

Section: An Lp-based Proof For the Squared Weight Algorithmmentioning

confidence: 99%

“…We are not aware of natural systems that are bit-concave but not concave. 5 The following lemma characterizes concave systems. A more general statement of this fact was proven by Calvillo [2].…”

Section: Examples Of Good Systemsmentioning

confidence: 99%

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Robust Randomized Matchings

Matuschke¹,

Skutella²,

Soto³

2018

Mathematics of OR

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The following game is played on a weighted graph: Alice selects a matching M and Bob selects a number k. Alice's payoff is the ratio of the weight of the k heaviest edges of M to the maximum weight of a matching of size at most k. If M guarantees a payoff of at least α then it is called α-robust. Hassin and Rubinstein [7] gave an algorithm that returns a 1/ √ 2-robust matching, which is best possible. We show that Alice can improve her payoff to 1/ ln(4) by playing a randomized strategy. This result extends to a very general class of independence systems that includes matroid intersection, b-matchings, and strong 2-exchange systems. It also implies an improved approximation factor for a stochastic optimization variant known as the maximum priority matching problem and translates to an asymptotic robustness guarantee for deterministic matchings, in which Bob can only select numbers larger than a given constant. Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound. * A preliminary version of this paper appeared in the proceedings of SODA 2015 [15].

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Section: Related Workmentioning

confidence: 85%

Section: An Lp-based Proof For the Squared Weight Algorithmmentioning

confidence: 99%

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Robust Randomized Matchings

Matuschke¹,

Skutella²,

Soto³

2018

Mathematics of OR

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“…Kakimura et al [26] proved that the deterministic version of the maximum cardinality robustness problem is weakly NP-hard but admits an FPTAS. Since Hassin and Rubinstein [23] introduced the notion of the cardinality robustness, many papers have been investigating the value of the maximum cardinality robustness [23,19,25]. Matuschke et al [37] introduced randomized strategies for the cardinality robustness, and they presented a randomized strategy with (1/ ln 4)-robustness for a certain class of independence system I. Kobayashi and Takazawa [31] focused on independence systems that are defined from the knapsack problem, and exhibited two randomized strategy with robustness Ω(1/ log σ) and Ω(1/ log υ), where σ is the exchangeability of the independence system and υ = the size of a maximum independent set the size of a minimum dependent set−1 .…”

Section: Related Workmentioning

confidence: 99%

“…Another example of (1) is to compute the cardinality robustness for the maximum weight independent set problem [23,19,25,37,31]. The problem is to choose an independent set of size at most k with as large total weight as possible, but the cardinality bound k is not known in advance.…”

Section: Introductionmentioning

confidence: 99%

Randomized Strategies for Robust Combinatorial Optimization

Kawase

Sumita

2019

AAAI

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In this paper, we study the following robust optimization problem. Given an independence system and candidate objective functions, we choose an independent set, and then an adversary chooses one objective function, knowing our choice. Our goal is to find a randomized strategy (i.e., a probability distribution over the independent sets) that maximizes the expected objective value. To solve the problem, we propose two types of schemes for designing approximation algorithms. One scheme is for the case when objective functions are linear. It first finds an approximately optimal aggregated strategy and then retrieves a desired solution with little loss of the objective value. The approximation ratio depends on a relaxation of an independence system polytope. As applications, we provide approximation algorithms for a knapsack constraint or a matroid intersection by developing appropriate relaxations and retrievals. The other scheme is based on the multiplicative weights update method. A key technique is to introduce a new concept called (η, γ)-reductions for objective functions with parameters η, γ. We show that our scheme outputs a nearly α-approximate solution if there exists an α-approximation algorithm for a subproblem defined by (η, γ)-reductions. This improves approximation ratio in previous results. Using our result, we provide approximation algorithms when the objective functions are submodular or correspond to the cardinality robustness for the knapsack problem.

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