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.

翻译：十年前，格哈德·韦格纳提出了一个被称为“韦格纳远足问题”的著名开放性问题：假设有 $n$ 人计划徒步旅行；每位成员以区间形式（介于 $1$ 到 $n$ 之间）表达其对徒步小组规模的偏好。能否高效地将这 $n$ 人分配到若干徒步子组中，使得每个人都认可其所分配子组的规模？该问题等价于高效判定一个具有区间认可偏好的匿名享乐博弈实例是否允许完美划分。我们通过提出一个时间复杂度为 $O(n^5)$ 的算法，肯定地解决了这一开放问题。我们的解决方案基于对计算几何中特定矩形穿刺问题采用动态规划方法。此外，我们提出了该问题自然且更具挑战性的扩展形式，例如最大化满意参与者数量及具有单峰偏好的变体，并证明这些扩展问题同样可高效求解。最后但同样重要的是，我们利用该解决方案高效计算了匿名单峰享乐博弈中最大化平等福利的划分。