Document

**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)

Copy BibTex To Clipboard

@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-dev.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} }

Document

**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)

Copy BibTex To Clipboard

@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-dev.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} }

X

Feedback for Dagstuhl Publishing

Feedback submitted

Please try again later or send an E-mail