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Solving Woeginger’s Hiking Problem: Wonderful Partitions in Anonymous Hedonic Games

Authors Andrei Constantinescu , Pascal Lenzner , Rebecca Reiffenhäuser , Daniel Schmand , Giovanna Varricchio

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Author Details

Andrei Constantinescu
  • ETH Zürich, Switzerland
Pascal Lenzner
  • Hasso Plattner Institute, University of Potsdam, Germany
Rebecca Reiffenhäuser
  • University of Amsterdam, The Netherlands
Daniel Schmand
  • University of Bremen, Germany
Giovanna Varricchio
  • University of Calabria, Rende, Italy

Funding

  • Varricchio, Giovanna: The author is grateful for the support by the PNRR MUR project FAIR - Future AI Research (PE00000013), and the DFG, German Research Foundation, grant (Ho 3831/5-1).

Acknowledgements

We would like to thank the Schloss Dagstuhl - Leibniz-Zentrum für Informatik who hosted the event "Computational Social Dynamics" (Seminar 22452) in November 2022 [Hoefer et al., 2023], where the work leading to this paper was started. We are grateful to Martin Hoefer, Sigal Oren, and Roger Wattenhofer for organizing this event. We additionally thank Roger Wattenhofer for the useful discussions concerning this work.

Cite AsGet BibTex

Andrei Constantinescu, Pascal Lenzner, Rebecca Reiffenhäuser, Daniel Schmand, and Giovanna Varricchio. Solving Woeginger’s Hiking Problem: Wonderful Partitions in Anonymous Hedonic Games. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.48

Abstract

A decade ago, Gerhard Woeginger posed an open problem that became well-known as "Woeginger’s Hiking Problem": Consider a group of n people that want to go hiking; everyone expresses preferences over the size of their hiking group in the form of an interval between 1 and n. Is it possible to efficiently assign the n people to a set of hiking subgroups so that every person approves the size of their assigned subgroup? The problem is also known as efficiently deciding if an instance of an anonymous Hedonic Game with interval approval preferences admits a wonderful partition. We resolve the open problem in the affirmative by presenting an O(n⁵) time algorithm for Woeginger’s Hiking Problem. Our solution is based on employing a dynamic programming approach for a specific rectangle stabbing problem from computational geometry. Moreover, we propose natural, more demanding extensions of the problem, e.g., maximizing the number of satisfied participants and variants with single-peaked preferences, and show that they are also efficiently solvable. Last but not least, we employ our solution to efficiently compute a partition that maximizes the egalitarian welfare for anonymous single-peaked Hedonic Games.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic programming
  • Theory of computation → Algorithmic game theory
  • Theory of computation → Discrete optimization
Keywords
  • Algorithmic Game Theory
  • Dynamic Programming
  • Anonymous Hedonic Games
  • Single-Peaked Preferences
  • Social Optimum
  • Wonderful Partitions

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