合作博弈的多面体方法：小提升与困难面

中文摘要 AI 辅助

我们研究了由合作博弈（也称为伪布尔函数）产生的多面体和锥体上的基本算法问题——成员测试、分离、有效不等式测试和线性优化——的计算复杂性。研究此类问题的一个核心障碍是，n个参与者的通用合作博弈需要2^n个值，因此对于具有n个参与者的博弈，输入大小为2^n，这使得这些计算任务在理论上变得平凡。限制为k-可加博弈可将输入大小减少到O(n^k)，使得此类博弈成为关于有效算法存在性问题的自然目标。在积极方面，我们给出了k-可加k-单调博弈的核心的显式扩展公式，大小为O(n^k)，允许所有四个问题通过单个多项式大小的线性规划解决——特别是绕过了从Deng和Papadimitriou或Edmonds的早期可处理性结果构建时所需的椭球体方法。对于k-可加(k-1)-单调博弈的锥体，我们给出了其极射线的完整刻画，并推导出扩展复杂度的相同O(n^k)界，从而提供了Billionnet和Minoux结果的基于几何的证明和推广。在消极方面，我们表明对于l≤k-2，k-可加l-单调博弈的锥体在计算上是难处理的：成员测试不在NP中（除非NP=coNP），有效不等式测试是NP完全的，扩展复杂度至少为1.5^n。我们的硬度结果作为特例，得到了Crama以及Gallo和Simone的结果。此外，我们的硬度结果也解释了k-可加(k-2)-单调博弈锥体的极射线缺乏良好刻画的原因。

英文摘要

We study the computational complexity of fundamental algorithmic problems -- membership testing, separation, valid-inequality testing, and linear optimization -- over polytopes and cones arising from cooperative games (also known as pseudo-Boolean functions). A central obstacle in the study of such problems is that a general cooperative game on $n$ players requires $2^n$ values, so the input size is $2^n$ for a game with $n$ players, making these computational tasks theoretically trivial. Restricting to $k$-additive games reduces the input size to $O(n^k)$, making such games a natural target for meaningful questions about the existence of efficient algorithms. On the positive side, we give an explicit extended formulation of size $O(n^k)$ for the core of $k$-additive $k$-monotone games, allowing all four problems to be solved by a single polynomial-size linear program -- in particular, circumventing the ellipsoid method that is needed when building from earlier tractability results of Deng and Papadimitriou, or of Edmonds. For the cone of $k$-additive $(k{-}1)$-monotone games, we give a complete characterization of its extreme rays and derive the same $O(n^k)$ bound on extension complexity, yielding a geometry-based proof and generalization of a result of Billionnet and Minoux. On the negative side, we show that for $l \leq k-2$ the cone of $k$-additive $l$-monotone games is computationally intractable: membership testing is not in NP (unless NP\,=\,coNP), valid-inequality testing is NP-complete, and extension complexity is at least $1.5^n$. Our hardness results yield, as a special case, a result of Crama and of Gallo and Simone. Furthermore, our hardness results also explain the lack of any good characterization of the extreme rays of the cone of $k$-additive $(k{-}2)$-monotone games.

发表机构

• Charles University, Faculty of Mathematics and Physics（查理大学数学与物理学院）
• Université Paris 1 Panthéon-Sorbonne, Centre d’Economie de la Sorbonne（巴黎第一先贤祠-索邦大学，索邦经济中心）

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