License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2017.36
URN: urn:nbn:de:0030-drops-81489
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/8148/
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Anari, Nima ; Oveis Gharan, Shayan ; Saberi, Amin ; Singh, Mohit

Nash Social Welfare, Matrix Permanent, and Stable Polynomials

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Abstract

We study the problem of allocating m items to n agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a 1/e approximation factor of the objective, breaking the 1/2e^(1/e) approximation factor of Cole and Gkatzelis.

Our main technical contribution is an extension of Gurvits's lower bound on the coefficient of the square-free monomial of a degree m-homogeneous stable polynomial on m variables to all homogeneous polynomials. We use this extension to analyze the expected welfare of the allocation returned by our randomized rounding algorithm.

BibTeX - Entry

@InProceedings{anari_et_al:LIPIcs:2017:8148,
  author =	{Nima Anari and Shayan Oveis Gharan and Amin Saberi and Mohit Singh},
  title =	{{Nash Social Welfare, Matrix Permanent, and Stable Polynomials}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{36:1--36:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Christos H. Papadimitriou},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8148},
  URN =		{urn:nbn:de:0030-drops-81489},
  doi =		{10.4230/LIPIcs.ITCS.2017.36},
  annote =	{Keywords: Nash Welfare, Permanent, Matching, Stable Polynomial, Randomized Algorithm, Saddle Point}
}

Keywords: Nash Welfare, Permanent, Matching, Stable Polynomial, Randomized Algorithm, Saddle Point
Collection: 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)
Issue Date: 2017
Date of publication: 28.11.2017

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