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

TL;DR

The paper resolves Woeginger's Hiking Problem by presenting a polynomial-time $O(n^5)$ dynamic-programming algorithm that finds a wonderful partition for interval-approval instances, via a reduction to capacitated rectangle stabbing. It extends the core method to handle variants such as minimizing the number of excluded hikers ($O(n^7)$) and maximizing the number of satisfied participants ($O(n^7\log n)$) and addresses weighted scenarios. The work further investigates single-peaked cost structures, achieving $O(n^5\log n)$ for naturally single-peaked costs and establishing a polynomial-time bridge to minimizing egalitarian welfare in anonymous Hedonic Games. Overall, the authors connect partition problems in anonymous hedonic games to geometric stabbing problems, providing a versatile DP framework and a comprehensive tractability map for related objectives.

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 $O(n^5)$ 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.

Figures (2)

• Figure 1: Gerhard Woeginger, Oberwolfach, 2011. photo_gerhard
• Figure 2: Examples of utility functions considered in this paper. (a) interval approval sets, (b) non-interval approval sets with two approved group sizes, (c) utility functions are single-peaked and all agents with the same peak have the same utility function (under some additional mild technical assumptions), (d) individual single-peaked utility functions for all agents (as shown, the functions may differ even if they have the same peak).

Theorems & Definitions (24)

• Lemma 0
• Lemma 0
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• Definition 8