Approximation and parameterized complexity of minimax approval voting

Marek Cygan; Łukasz Kowalik; Arkadiusz Socała; Krzysztof Sornat

Approximation and parameterized complexity of minimax approval voting

Marek Cygan, Łukasz Kowalik, Arkadiusz Socała, Krzysztof Sornat

Research output: Contribution to conference › Paper › peer-review

7 Scopus citations

Abstract

We present three results on the complexity of MINIMAX APPROVAL VOTING. First, we study MINIMAX APPROVAL VOTING parameterized by the Hamming distance d from the solution to the votes. We show MINIMAX APPROVAL VOTING admits no algorithm running in time O^&z.ast; (2^{o(d log d})), unless the Exponential Time Hypothesis (ETH) fails. This means that the O^&z.ast;(d^2d) algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time O^&z.ast;((3/e)²), which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for MINIMAX APPROVAL VOTING, which runs intimenO(^1/e2·^log(^1/ϵ)) · poly (m), almost matching the running time of the fastest known PTAS for CLOSEST STRING due to Ma and Sun [SIAM J. Comp. 2009].

Original language	English
Pages	459-465
Number of pages	7
State	Published - 1 Jan 2017
Externally published	Yes
Event	31st AAAI Conference on Artificial Intelligence, AAAI 2017 - San Francisco, United States Duration: 4 Feb 2017 → 10 Feb 2017

Conference

Conference	31st AAAI Conference on Artificial Intelligence, AAAI 2017
Country/Territory	United States
City	San Francisco
Period	4/02/17 → 10/02/17

ASJC Scopus subject areas

Artificial Intelligence

Cite this

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title = "Approximation and parameterized complexity of minimax approval voting",

abstract = "We present three results on the complexity of MINIMAX APPROVAL VOTING. First, we study MINIMAX APPROVAL VOTING parameterized by the Hamming distance d from the solution to the votes. We show MINIMAX APPROVAL VOTING admits no algorithm running in time O&z.ast; (2o(d log d)), unless the Exponential Time Hypothesis (ETH) fails. This means that the O&z.ast;(d2d) algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time O&z.ast;((3/e)2), which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for MINIMAX APPROVAL VOTING, which runs intimenO(1/e2·log(1/ϵ)) · poly (m), almost matching the running time of the fastest known PTAS for CLOSEST STRING due to Ma and Sun [SIAM J. Comp. 2009].",

author = "Marek Cygan and {\L}ukasz Kowalik and Arkadiusz Soca{\l}a and Krzysztof Sornat",

note = "Publisher Copyright: Copyright {\textcopyright} 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.; 31st AAAI Conference on Artificial Intelligence, AAAI 2017 ; Conference date: 04-02-2017 Through 10-02-2017",

year = "2017",

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language = "English",

pages = "459--465",

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T1 - Approximation and parameterized complexity of minimax approval voting

AU - Cygan, Marek

AU - Kowalik, Łukasz

AU - Socała, Arkadiusz

AU - Sornat, Krzysztof

PY - 2017/1/1

Y1 - 2017/1/1

N2 - We present three results on the complexity of MINIMAX APPROVAL VOTING. First, we study MINIMAX APPROVAL VOTING parameterized by the Hamming distance d from the solution to the votes. We show MINIMAX APPROVAL VOTING admits no algorithm running in time O&z.ast; (2o(d log d)), unless the Exponential Time Hypothesis (ETH) fails. This means that the O&z.ast;(d2d) algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time O&z.ast;((3/e)2), which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for MINIMAX APPROVAL VOTING, which runs intimenO(1/e2·log(1/ϵ)) · poly (m), almost matching the running time of the fastest known PTAS for CLOSEST STRING due to Ma and Sun [SIAM J. Comp. 2009].

AB - We present three results on the complexity of MINIMAX APPROVAL VOTING. First, we study MINIMAX APPROVAL VOTING parameterized by the Hamming distance d from the solution to the votes. We show MINIMAX APPROVAL VOTING admits no algorithm running in time O&z.ast; (2o(d log d)), unless the Exponential Time Hypothesis (ETH) fails. This means that the O&z.ast;(d2d) algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time O&z.ast;((3/e)2), which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for MINIMAX APPROVAL VOTING, which runs intimenO(1/e2·log(1/ϵ)) · poly (m), almost matching the running time of the fastest known PTAS for CLOSEST STRING due to Ma and Sun [SIAM J. Comp. 2009].

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M3 - Paper

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T2 - 31st AAAI Conference on Artificial Intelligence, AAAI 2017

Y2 - 4 February 2017 through 10 February 2017

ER -

Approximation and parameterized complexity of minimax approval voting

Abstract

Conference

ASJC Scopus subject areas

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