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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

In cooperative game theory, the primary focus is the equitable allocation of payoffs or costs among agents. However, in the practical applications of cooperative games, accurately representing games is challenging. In such cases, using an allocation method sensitive to small perturbations in the game can lead to various problems, including dissatisfaction among agents and the potential for manipulation by agents seeking to maximize their own benefits. Therefore, the allocation method must be robust against game perturbations.
In this study, we explore optimization games, in which the value of the characteristic function is provided as the optimal value of an optimization problem. To assess the robustness of the allocation methods, we use the Lipschitz constant, which quantifies the extent of change in the allocation vector in response to a unit perturbation in the weight vector of the underlying problem. Thereafter, we provide an algorithm for the matching game that returns an allocation belonging to the (1/2-ε)-approximate core with Lipschitz constant O(ε^{-1}). Additionally, we provide an algorithm for a minimum spanning tree game that returns an allocation belonging to the 4-approximate core with a constant Lipschitz constant.
The Shapley value is a popular allocation that satisfies several desirable properties. Therefore, we investigate the robustness of the Shapley value. We demonstrate that the Lipschitz constant of the Shapley value for the minimum spanning tree is constant, whereas that for the matching game is Ω(log n), where n denotes the number of vertices.

Soh Kumabe and Yuichi Yoshida. Lipschitz Continuous Allocations for Optimization Games. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 102:1-102:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kumabe_et_al:LIPIcs.ICALP.2024.102, author = {Kumabe, Soh and Yoshida, Yuichi}, title = {{Lipschitz Continuous Allocations for Optimization Games}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {102:1--102:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.102}, URN = {urn:nbn:de:0030-drops-202456}, doi = {10.4230/LIPIcs.ICALP.2024.102}, annote = {Keywords: Cooperative Games, Lipschitz Continuity} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

In resource allocation, we often require that the output allocation of an algorithm is stable against input perturbation because frequent reallocation is costly and untrustworthy. Varma and Yoshida (SODA'21) formalized this requirement for algorithms as the notion of average sensitivity. Here, the average sensitivity of an algorithm on an input instance is, roughly speaking, the average size of the symmetric difference of the output for the instance and that for the instance with one item deleted, where the average is taken over the deleted item.
In this work, we consider the average sensitivity of the knapsack problem, a representative example of a resource allocation problem. We first show a (1-ε)-approximation algorithm for the knapsack problem with average sensitivity O(ε^{-1}log ε^{-1}). Then, we complement this result by showing that any (1-ε)-approximation algorithm has average sensitivity Ω(ε^{-1}). As an application of our algorithm, we consider the incremental knapsack problem in the random-order setting, where the goal is to maintain a good solution while items arrive one by one in a random order. Specifically, we show that for any ε > 0, there exists a (1-ε)-approximation algorithm with amortized recourse O(ε^{-1}log ε^{-1}) and amortized update time O(log n+f_ε), where n is the total number of items and f_ε > 0 is a value depending on ε.

Soh Kumabe and Yuichi Yoshida. Average Sensitivity of the Knapsack Problem. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 75:1-75:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kumabe_et_al:LIPIcs.ESA.2022.75, author = {Kumabe, Soh and Yoshida, Yuichi}, title = {{Average Sensitivity of the Knapsack Problem}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {75:1--75:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.75}, URN = {urn:nbn:de:0030-drops-170136}, doi = {10.4230/LIPIcs.ESA.2022.75}, annote = {Keywords: Average Sensitivity, Knapsack Problem, FPRAS} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

In this paper, we introduce the interval query problem on cube-free median graphs. Let G be a cube-free median graph and 𝒮 be a commutative semigroup. For each vertex v in G, we are given an element p(v) in 𝒮. For each query, we are given two vertices u,v in G and asked to calculate the sum of p(z) over all vertices z belonging to a u-v shortest path. This is a common generalization of range query problems on trees and grids. In this paper, we provide an algorithm to answer each interval query in O(log² n) time. The required data structure is constructed in O(n log³ n) time and O(n log² n) space. To obtain our algorithm, we introduce a new technique, named the staircases decomposition, to decompose an interval of cube-free median graphs into simpler substructures.

Soh Kumabe. Interval Query Problem on Cube-Free Median Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kumabe:LIPIcs.ISAAC.2021.18, author = {Kumabe, Soh}, title = {{Interval Query Problem on Cube-Free Median Graphs}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.18}, URN = {urn:nbn:de:0030-drops-154510}, doi = {10.4230/LIPIcs.ISAAC.2021.18}, annote = {Keywords: Data Structures, Range Query Problems, Median Graphs} }

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