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DOI: 10.4230/LIPIcs.ISAAC.2021.49

Making Three out of Two: Three-Way Online Correlated Selection

Authors: Yongho Shin and Hyung-Chan An

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

Abstract

Two-way online correlated selection (two-way OCS) is an online algorithm that, at each timestep, takes a pair of elements from the ground set and irrevocably chooses one of the two elements, while ensuring negative correlation in the algorithm's choices. Whilst OCS was initially invented by Fahrbach, Huang, Tao, and Zadimoghaddam to break a natural long-standing barrier in the edge-weighted online bipartite matching problem, it is an interesting technique on its own due to its capability of introducing a powerful algorithmic tool, namely negative correlation, to online algorithms. As such, Fahrbach et al. posed two tantalizing open questions in their paper, one of which was the following: Can we obtain n-way OCS for n > 2, in which the algorithm can be given n > 2 elements to choose from at each timestep? In this paper, we affirmatively answer this open question by presenting a three-way OCS. Our algorithm uses two-way OCS as its building block and is simple to describe; however, as it internally runs two instances of two-way OCS, one of which is fed with the output of the other, the final output probability distribution becomes highly elusive. We tackle this difficulty by approximating the output distribution of OCS by a flat, less correlated function and using it as a safe "surrogate" of the real distribution. Our three-way OCS also yields a 0.5093-competitive algorithm for edge-weighted online matching, demonstrating its usefulness.

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Yongho Shin and Hyung-Chan An. Making Three out of Two: Three-Way Online Correlated Selection. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 49:1-49:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{shin_et_al:LIPIcs.ISAAC.2021.49,
  author =	{Shin, Yongho and An, Hyung-Chan},
  title =	{{Making Three out of Two: Three-Way Online Correlated Selection}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{49:1--49:17},
  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.2021.49},
  URN =		{urn:nbn:de:0030-drops-154822},
  doi =		{10.4230/LIPIcs.ISAAC.2021.49},
  annote =	{Keywords: online correlated selection, multi-way OCS, online algorithms, negative correlation, edge-weighted online bipartite matching}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2020.21

Constant-Factor Approximation Algorithms for the Parity-Constrained Facility Location Problem

Authors: Kangsan Kim, Yongho Shin, and Hyung-Chan An

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract

Facility location is a prominent optimization problem that has inspired a large quantity of both theoretical and practical studies in combinatorial optimization. Although the problem has been investigated under various settings reflecting typical structures within the optimization problems of practical interest, little is known on how the problem behaves in conjunction with parity constraints. This shortfall of understanding was rather discouraging when we consider the central role of parity in the field of combinatorics. In this paper, we present the first constant-factor approximation algorithm for the facility location problem with parity constraints. We are given as the input a metric on a set of facilities and clients, the opening cost of each facility, and the parity requirement - odd, even, or unconstrained - of every facility in this problem. The objective is to open a subset of facilities and assign every client to an open facility so as to minimize the sum of the total opening costs and the assignment distances, but subject to the condition that the number of clients assigned to each open facility must have the same parity as its requirement. Although the unconstrained facility location problem as a relaxation for this parity-constrained generalization has unbounded gap, we demonstrate that it yields a structured solution whose parity violation can be corrected at small cost. This correction is prescribed by a T-join on an auxiliary graph constructed by the algorithm. This auxiliary graph does not satisfy the triangle inequality, but we show that a carefully chosen set of shortcutting operations leads to a cheap and sparse T-join. Finally, we bound the correction cost by exhibiting a combinatorial multi-step construction of an upper bound.

Cite as

Kangsan Kim, Yongho Shin, and Hyung-Chan An. Constant-Factor Approximation Algorithms for the Parity-Constrained Facility Location Problem. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kim_et_al:LIPIcs.ISAAC.2020.21,
  author =	{Kim, Kangsan and Shin, Yongho and An, Hyung-Chan},
  title =	{{Constant-Factor Approximation Algorithms for the Parity-Constrained Facility Location Problem}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{21:1--21:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.21},
  URN =		{urn:nbn:de:0030-drops-133652},
  doi =		{10.4230/LIPIcs.ISAAC.2020.21},
  annote =	{Keywords: Facility location problems, approximation algorithms, clustering problems, parity constraints}
}

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Documents authored by Shin, Yongho

Making Three out of Two: Three-Way Online Correlated Selection

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Constant-Factor Approximation Algorithms for the Parity-Constrained Facility Location Problem

Abstract

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