Computing Knapsack Solutions with Cardinality Robustness

Kakimura, Naonori; Makino, Kazuhisa; Seimi, Kento

doi:10.1007/978-3-642-25591-5_71

“…Hassin and Segev [8] considered the problem of finding a small subgraph of a given graph that contains for every k ∈ N a spanning tree (or path, respectively) of cardinality at most k and weight at least α times the weight of a maximum weight solution of size k. They show that α|V |/(1 − α 2 ) edges suffice and give polynomial time algorithms for robust solutions using the results in [7]. In a wider context, Hassin and Rubinstein's work inspired other robustness results, e. g., in graph coloring [6], knapsack problems [12,3], and sequencing [16]. More recently, Dütting, Roughgarden, and Talgam-Cohen [4] discovered an application of Hassin and Rubinstein's analysis of the squared weight algorithm in the design of double auctions.…”

Section: Related Workmentioning

confidence: 99%

Robust randomized matchings

Matuschke¹,

Skutella²,

Soto³

2014

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

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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.

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“…Hassin and Segev [8] considered the problem of finding a small subgraph of a given graph that contains for every k ∈ N a spanning tree (or path, respectively) of cardinality at most k and weight at least α times the weight of a maximum weight solution of size k. They show that α|V |/(1 − α 2 ) edges suffice and give polynomial time algorithms for robust solutions using the results in [7]. In a wider context, Hassin and Rubinstein's work inspired other robustness results, e. g., in graph coloring [6], knapsack problems [12,3], and sequencing [16]. More recently, Dütting, Roughgarden, and Talgam-Cohen [4] discovered an application of Hassin and Rubinstein's analysis of the squared weight algorithm in the design of double auctions.…”

Section: Related Workmentioning

confidence: 99%

Robust Randomized Matchings

Matuschke¹,

Skutella²,

Soto³

2018

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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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“…In [6], it is also shown that the above robustness is tight in the sense that for an arbitrary positive integer µ, there exists an independence system (E, F) such that µ(F) = µ and no α-robust solution exists for arbitrary α > 1/ √ µ. Kakimura, Makino, and Seimi [7] focused on the case where (E, F) is defined by an instance of the knapsack problem. An instance (E, p, w, C) of the knapsack problem consists of the set E of items, the profit vector p ∈ R E + , the weight vector w ∈ R E + , and the capacity C ∈ R + .…”

Section: Cardinality Robustness In Independence Systemsmentioning

confidence: 99%

“…A subset X ⊆ E is feasible if its weight w(X) := ∑ e∈X w e is at most the capacity, i.e., F = {X ⊆ E | w(X) ≤ C}. Kakimura, Makino, and Seimi [7] proved that the problem of computing a knapsack solution with the maximum robustness is weakly NP-hard, and also presented a fully polynomial-time approximation scheme (FPTAS) for this problem.…”

Section: Cardinality Robustness In Independence Systemsmentioning

confidence: 99%

Randomized strategies for cardinality robustness in the knapsack problem

Kobayashi

¹

,

Takazawa

²

2017

Theoretical Computer Science

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We consider the following zero-sum game related to the knapsack problem. Given an instance of the knapsack problem, Alice chooses a knapsack solution and Bob, knowing Alice's solution, chooses a cardinality k. Then, Alice obtains a payoff equal to the ratio of the profit of the best k items in her solution to that of the best solution of size at most k. For α > 0, a knapsack solution is called α-robust if it guarantees payoff α. If Alice adopts a deterministic strategy, the objective of Alice is to find a max-robust knapsack solution. By applying the argument in Kakimura and Makino (2013) for robustness in general independence systems, a (1/ √ µ)-robust solution exists and is found in polynomial time,

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Computing Knapsack Solutions with Cardinality Robustness

Cited by 11 publications

References 16 publications

Robust randomized matchings

Robust randomized matchings

Robust Randomized Matchings

Randomized strategies for cardinality robustness in the knapsack problem

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