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DOI: 10.4230/LIPIcs.ITCS.2026.69

Computing Equilibrium Points of Electrostatic Potentials

Authors: Abheek Ghosh, Paul W. Goldberg, and Alexandros Hollender

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We study the computation of equilibrium points of electrostatic potentials: locations in space where the electrostatic force arising from a collection of charged particles vanishes. This is a novel scenario of optimization in which solutions are guaranteed to exist due to a nonconstructive argument, but gradient descent is unreliable due to the presence of singularities. We present an algorithm based on piecewise approximation of the potential function by Taylor series. The main insight is to divide the domain into a grid with variable coarseness, where grid cells are exponentially smaller in regions where the function changes rapidly compared to regions where it changes slowly. Our algorithm finds approximate equilibrium points in time poly-logarithmic in the approximation parameter, but these points are not guaranteed to be close to exact solutions. Nevertheless, we show that such points can be computed efficiently under a mild assumption that we call "strong non-degeneracy". We complement these algorithmic results by studying a generalization of this problem and showing that it is CLS-hard and in PPAD, leaving its precise classification as an intriguing open problem.

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Abheek Ghosh, Paul W. Goldberg, and Alexandros Hollender. Computing Equilibrium Points of Electrostatic Potentials. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 69:1-69:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ghosh_et_al:LIPIcs.ITCS.2026.69,
  author =	{Ghosh, Abheek and Goldberg, Paul W. and Hollender, Alexandros},
  title =	{{Computing Equilibrium Points of Electrostatic Potentials}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{69:1--69:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.69},
  URN =		{urn:nbn:de:0030-drops-253566},
  doi =		{10.4230/LIPIcs.ITCS.2026.69},
  annote =	{Keywords: Total search problems, TFNP, PPAD, CLS, polynomial equations}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2018.27

On the Welfare of Cardinal Voting Mechanisms

Authors: Umang Bhaskar and Abheek Ghosh

Published in: LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

Abstract

A voting mechanism is a method for preference aggregation that takes as input preferences over alternatives from voters, and selects an alternative, or a distribution over alternatives. While preferences of voters are generally assumed to be cardinal utility functions that map each alternative to a real value, mechanisms typically studied assume coarser inputs, such as rankings of the alternatives (called ordinal mechanisms). We study cardinal mechanisms, that take as input the cardinal utilities of the voters, with the objective of minimizing the distortion - the worst-case ratio of the best social welfare to that obtained by the mechanism. For truthful cardinal mechanisms with m alternatives and n voters, we show bounds of Theta(mn), Omega(m), and Omega(sqrt{m}) for deterministic, unanimous, and randomized mechanisms respectively. This shows, somewhat surprisingly, that even mechanisms that allow cardinal inputs have large distortion. There exist ordinal (and hence, cardinal) mechanisms with distortion O(sqrt{m log m}), and hence our lower bound for randomized mechanisms is nearly tight. In an effort to close this gap, we give a class of truthful cardinal mechanisms that we call randomized hyperspherical mechanisms that have O(sqrt{m log m}) distortion. These are interesting because they violate two properties - localization and non-perversity - that characterize truthful ordinal mechanisms, demonstrating non-trivial mechanisms that differ significantly from ordinal mechanisms. Given the strong lower bounds for truthful mechanisms, we then consider approximately truthful mechanisms. We give a mechanism that is delta-truthful given delta in (0,1), and has distortion close to 1. Finally, we consider the simple mechanism that selects the alternative that maximizes social welfare. This mechanism is not truthful, and we study the distortion at equilibria for the voters (equivalent to the Price of Anarchy, or PoA). While in general, the PoA is unbounded, we show that for equilibria obtained from natural dynamics, the PoA is close to 1. Thus relaxing the notion of truthfulness in both cases allows us to obtain near-optimal distortion.

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Umang Bhaskar and Abheek Ghosh. On the Welfare of Cardinal Voting Mechanisms. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 27:1-27:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bhaskar_et_al:LIPIcs.FSTTCS.2018.27,
  author =	{Bhaskar, Umang and Ghosh, Abheek},
  title =	{{On the Welfare of Cardinal Voting Mechanisms}},
  booktitle =	{38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)},
  pages =	{27:1--27:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-093-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{122},
  editor =	{Ganguly, Sumit and Pandya, Paritosh},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.27},
  URN =		{urn:nbn:de:0030-drops-99260},
  doi =		{10.4230/LIPIcs.FSTTCS.2018.27},
  annote =	{Keywords: computational social choice, voting rules, cardinal mechanisms, price of anarchy, distortion}
}

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Documents authored by Ghosh, Abheek

Computing Equilibrium Points of Electrostatic Potentials

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On the Welfare of Cardinal Voting Mechanisms

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