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Documents authored by Krishna, Anand

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DOI: 10.4230/LIPIcs.ITCS.2024.8
Sublinear Approximation Algorithm for Nash Social Welfare with XOS Valuations

Authors: Siddharth Barman, Anand Krishna, Pooja Kulkarni, and Shivika Narang

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
We study the problem of allocating indivisible goods among n agents with the objective of maximizing Nash social welfare (NSW). This welfare function is defined as the geometric mean of the agents' valuations and, hence, it strikes a balance between the extremes of social welfare (arithmetic mean) and egalitarian welfare (max-min value). Nash social welfare has been extensively studied in recent years for various valuation classes. In particular, a notable negative result is known when the agents' valuations are complement-free and are specified via value queries: for XOS valuations, one necessarily requires exponentially many value queries to find any sublinear (in n) approximation for NSW. Indeed, this lower bound implies that stronger query models are needed for finding better approximations. Towards this, we utilize demand oracles and XOS oracles; both of these query models are standard and have been used in prior work on social welfare maximization with XOS valuations. We develop the first sublinear approximation algorithm for maximizing Nash social welfare under XOS valuations, specified via demand and XOS oracles. Hence, this work breaks the O(n)-approximation barrier for NSW maximization under XOS valuations. We obtain this result by developing a novel connection between NSW and social welfare under a capped version of the agents' valuations. In addition to this insight, which might be of independent interest, this work relies on an intricate combination of multiple technical ideas, including the use of repeated matchings and the discrete moving knife method. In addition, we partially complement the algorithmic result by showing that, under XOS valuations, an exponential number of demand and XOS queries are necessarily required to approximate NSW within a factor of (1 - 1/e).
Cite as

Siddharth Barman, Anand Krishna, Pooja Kulkarni, and Shivika Narang. Sublinear Approximation Algorithm for Nash Social Welfare with XOS Valuations. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 8:1-8:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{barman_et_al:LIPIcs.ITCS.2024.8,
  author =	{Barman, Siddharth and Krishna, Anand and Kulkarni, Pooja and Narang, Shivika},
  title =	{{Sublinear Approximation Algorithm for Nash Social Welfare with XOS Valuations}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{8:1--8:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.8},
  URN =		{urn:nbn:de:0030-drops-195366},
  doi =		{10.4230/LIPIcs.ITCS.2024.8},
  annote =	{Keywords: Discrete Fair Division, Nash Social Welfare, XOS Valuations}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.11
Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations

Authors: Siddharth Barman, Umang Bhaskar, Anand Krishna, and Ranjani G. Sundaram

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
We develop polynomial-time algorithms for the fair and efficient allocation of indivisible goods among n agents that have subadditive valuations over the goods. We first consider the Nash social welfare as our objective and design a polynomial-time algorithm that, in the value oracle model, finds an 8n-approximation to the Nash optimal allocation. Subadditive valuations include XOS (fractionally subadditive) and submodular valuations as special cases. Our result, even for the special case of submodular valuations, improves upon the previously best known O(n log n)-approximation ratio of Garg et al. (2020). More generally, we study maximization of p-mean welfare. The p-mean welfare is parameterized by an exponent term p ∈ (-∞, 1] and encompasses a range of welfare functions, such as social welfare (p = 1), Nash social welfare (p → 0), and egalitarian welfare (p → -∞). We give an algorithm that, for subadditive valuations and any given p ∈ (-∞, 1], computes (in the value oracle model and in polynomial time) an allocation with p-mean welfare at least 1/(8n) times the optimal. Further, we show that our approximation guarantees are essentially tight for XOS and, hence, subadditive valuations. We adapt a result of Dobzinski et al. (2010) to show that, under XOS valuations, an O (n^{1-ε}) approximation for the p-mean welfare for any p ∈ (-∞,1] (including the Nash social welfare) requires exponentially many value queries; here, ε > 0 is any fixed constant.
Cite as

Siddharth Barman, Umang Bhaskar, Anand Krishna, and Ranjani G. Sundaram. Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{barman_et_al:LIPIcs.ESA.2020.11,
  author =	{Barman, Siddharth and Bhaskar, Umang and Krishna, Anand and Sundaram, Ranjani G.},
  title =	{{Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{11:1--11:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.11},
  URN =		{urn:nbn:de:0030-drops-128775},
  doi =		{10.4230/LIPIcs.ESA.2020.11},
  annote =	{Keywords: Discrete Fair Division, Nash Social Welfare, Subadditive Valuations, Submodular Valuations}
}
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