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.
@InProceedings{anari_et_al:LIPIcs.ITCS.2017.36, author = {Anari, Nima and Oveis Gharan, Shayan and Saberi, Amin and Singh, Mohit}, 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 = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.36}, 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} }
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