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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2026.90

New Convex Programming Technique for Nash Social Welfare and Scheduling

Authors: Yuda Feng, Weijiang Hu, and Shi Li

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract

We propose a new convex programming relaxation for the weighted Nash social welfare (NSW) problem that achieves a matching (e^{1/e} ≈ 1.445)-approximation via the rounding algorithm of Feng and Li. Unlike the exponential-size configuration LP used in prior work, our formulation can be converted into a compact linear program of polynomial size, incurring only an additive loss of ln(1+ε) in the objective. This allows the program to be solved directly using standard LP solvers, without the ellipsoid method or dual separation oracles. In the unweighted case, we show that our convex program is equivalent to the restricted-spending Fisher market convex program of Cole and Gkatzelis, yielding a constructive proof that its integrality gap is exactly e^{1/e}. With a minor modification, our analysis also gives a simple proof of the e^{1/e} EF1 gap for the identical agent setting. Finally, we show that our convex programming technique extends to two unrelated machine scheduling problems, recovering the best-known approximation ratios with simpler analyses.

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Yuda Feng, Weijiang Hu, and Shi Li. New Convex Programming Technique for Nash Social Welfare and Scheduling. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 90:1-90:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{feng_et_al:LIPIcs.ICALP.2026.90,
  author =	{Feng, Yuda and Hu, Weijiang and Li, Shi},
  title =	{{New Convex Programming Technique for Nash Social Welfare and Scheduling}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{90:1--90:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.90},
  URN =		{urn:nbn:de:0030-drops-264797},
  doi =		{10.4230/LIPIcs.ICALP.2026.90},
  annote =	{Keywords: Nash Social Welfare, Convex Programming, Approximation Algorithms, Scheduling}
}

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New Convex Programming Technique for Nash Social Welfare and Scheduling

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