[论文解读] Coalition Games on Interaction Graphs: A Horticultural Perspective

Nicolás Bousquet, Zhentao Li+1 more|arXiv (Cornell University)|Feb 26, 2015

Game Theory and Voting Systems参考文献 14被引用 6

一句话总结

本文引入了丛参数（thicket parameter）以精确刻画图形联盟博弈中打包-覆盖比与整数规划间隙，其中联盟仅在诱导出连通子图时才有效。本文证明丛数等于最小宽度的藤蔓分解（vine decomposition），并为原始与对偶整数规划间隙提供了紧致界，表明原始间隙与丛数呈线性关系，而对偶间隙则为多项式关系。

ABSTRACT

We examine cooperative games where the viability of a coalition is determined by whether or not its members have the ability to communicate amongst themselves independently of non-members. This necessary condition for viability was proposed by Myerson (1977) and is modeled via an interaction graph $G=(V,E)$; a coalition $S\subseteq V$ is then viable if and only if the induced graph $G[S]$ is connected. The non-emptiness of the core of a coalition game can be tested by a well-known covering LP. Moreover, the integrality gap of its dual packing LP defines exactly the multiplicative least-core and the relative cost of stability of the coalition game. This gap is upper bounded by the packing-covering ratio which, for graphical coalition games, is known to be at most the treewidth of the interaction graph plus one (Meir et al. 2013). We examine the packing-covering ratio and integrality gaps of graphical coalition games in more detail. We introduce the thicket parameter of a graph, and prove it precisely measures the packing-covering ratio. It also approximately measures the primal and dual integrality gaps. The thicket number provides an upper bound of both integrality gaps. Moreover we show that for any interaction graph, the primal integrality gap is, in the worst case, linear in terms of the thicket number while the dual integrality gap is polynomial in terms of it. At the heart of our results, is a graph theoretic minmax theorem showing the thicket number is equal to the minimum width of a vine decomposition of the coalition graph (a vine decomposition is a generalization of a tree decomposition). We also explain how the thicket number relates to the VC-dimension of the set system produced by the game.

研究动机与目标

• 理解在依赖于交互图连通性的图形联盟博弈中的整数规划间隙。
• 通过一种新的图参数刻画乘法最小核心与稳定性相对成本。
• 以一种新颖的图论度量为基础，建立原始与对偶整数规划间隙的紧致界。
• 将丛参数与现有概念（如树宽与集合系统中的VC维）相联系。

提出的方法

• 将丛参数引入为图的藤蔓分解最小宽度的度量。
• 证明丛数等于图形联盟博弈中打包-覆盖比。
• 利用极小-极大定理，将丛数等同于藤蔓分解的最小宽度。
• 在路径的幂图上构造一族博弈，以建立原始整数规划间隙的紧致下界。
• 应用覆盖与打包线性规划之间的强对偶性，将最小核心与对偶整数规划间隙关联。
• 证明原始整数规划间隙至多与丛数呈线性关系，而对偶间隙则为丛数的多项式函数。

实验结果

研究问题

• RQ1 何种图参数能精确度量图形联盟博弈中的打包-覆盖比？
• RQ2 原始与对偶整数规划间隙如何与交互图的结构相关联？
• RQ3 丛数能否通过涉及藤蔓分解的极小-极大定理来刻画？
• RQ4 以丛数表示时，原始整数规划间隙的最紧可能上界为何？
• RQ5 丛数如何与树宽、VC维等其他图参数相关联？

主要发现

• 丛数在图形联盟博弈中恰好等于打包-覆盖比。
• 丛数等于交互图藤蔓分解的最小宽度。
• 原始整数规划间隙至多与丛数呈线性关系，且对路径的幂图而言该界是紧致的。
• 对偶整数规划间隙为丛数的多项式函数，其下界形式为 (1−2/k)·τ(G)，当 n 较大时成立。
• 丛数为原始与对偶整数规划间隙均提供了上界。
• 丛数与博弈中有效联盟所诱导的集合系统的VC维相关。