Improved Integrality Gap in Max-Min Allocation: or Topology at the North Pole
In the max-min allocation problem a set $P$ of players are to be allocated disjoint subsets of a set $R$ of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsi...
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ftdatacite:10.48550/arxiv.2202.01143 2023-05-15T17:39:57+02:00 Improved Integrality Gap in Max-Min Allocation: or Topology at the North Pole Haxell, Penny Szabó, Tibor 2022 https://dx.doi.org/10.48550/arxiv.2202.01143 https://arxiv.org/abs/2202.01143 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Data Structures and Algorithms cs.DS Discrete Mathematics cs.DM FOS Computer and information sciences Article CreativeWork article Preprint 2022 ftdatacite https://doi.org/10.48550/arxiv.2202.01143 2022-03-10T10:41:09Z In the max-min allocation problem a set $P$ of players are to be allocated disjoint subsets of a set $R$ of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bezáková and Dani showed that this problem is NP-hard to approximate within a factor less than $2$, consequently a great deal of work has focused on approximate solutions. The principal approach for obtaining approximation algorithms has been via the Configuration LP (CLP) of Bansal and Sviridenko. Accordingly, there has been much interest in bounding the integrality gap of this CLP. The existing algorithms and integrality gap estimations are all based one way or another on the combinatorial augmenting tree argument of Haxell for finding perfect matchings in certain hypergraphs. Our main innovation in this paper is to introduce the use of topological methods for the restricted max-min allocation problem, to replace the combinatorial argument. This approach yields substantial improvements in the integrality gap of the CLP. In particular we improve the previously best known bound of $3.808$ to $3.534$. We also study the $(1,\varepsilon)$-restricted version, in which resources can take only two values, and improve the integrality gap in most cases. : This is the full version of our paper submitted to STOC Article in Journal/Newspaper North Pole DataCite Metadata Store (German National Library of Science and Technology) North Pole |
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DataCite Metadata Store (German National Library of Science and Technology) |
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Data Structures and Algorithms cs.DS Discrete Mathematics cs.DM FOS Computer and information sciences |
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Data Structures and Algorithms cs.DS Discrete Mathematics cs.DM FOS Computer and information sciences Haxell, Penny Szabó, Tibor Improved Integrality Gap in Max-Min Allocation: or Topology at the North Pole |
topic_facet |
Data Structures and Algorithms cs.DS Discrete Mathematics cs.DM FOS Computer and information sciences |
description |
In the max-min allocation problem a set $P$ of players are to be allocated disjoint subsets of a set $R$ of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bezáková and Dani showed that this problem is NP-hard to approximate within a factor less than $2$, consequently a great deal of work has focused on approximate solutions. The principal approach for obtaining approximation algorithms has been via the Configuration LP (CLP) of Bansal and Sviridenko. Accordingly, there has been much interest in bounding the integrality gap of this CLP. The existing algorithms and integrality gap estimations are all based one way or another on the combinatorial augmenting tree argument of Haxell for finding perfect matchings in certain hypergraphs. Our main innovation in this paper is to introduce the use of topological methods for the restricted max-min allocation problem, to replace the combinatorial argument. This approach yields substantial improvements in the integrality gap of the CLP. In particular we improve the previously best known bound of $3.808$ to $3.534$. We also study the $(1,\varepsilon)$-restricted version, in which resources can take only two values, and improve the integrality gap in most cases. : This is the full version of our paper submitted to STOC |
format |
Article in Journal/Newspaper |
author |
Haxell, Penny Szabó, Tibor |
author_facet |
Haxell, Penny Szabó, Tibor |
author_sort |
Haxell, Penny |
title |
Improved Integrality Gap in Max-Min Allocation: or Topology at the North Pole |
title_short |
Improved Integrality Gap in Max-Min Allocation: or Topology at the North Pole |
title_full |
Improved Integrality Gap in Max-Min Allocation: or Topology at the North Pole |
title_fullStr |
Improved Integrality Gap in Max-Min Allocation: or Topology at the North Pole |
title_full_unstemmed |
Improved Integrality Gap in Max-Min Allocation: or Topology at the North Pole |
title_sort |
improved integrality gap in max-min allocation: or topology at the north pole |
publisher |
arXiv |
publishDate |
2022 |
url |
https://dx.doi.org/10.48550/arxiv.2202.01143 https://arxiv.org/abs/2202.01143 |
geographic |
North Pole |
geographic_facet |
North Pole |
genre |
North Pole |
genre_facet |
North Pole |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.2202.01143 |
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1766140719967240192 |