[论文解读] Graph Operations on Parity Games and Polynomial-Time Algorithms

Christoph Dittmann, Stephan Kreutzer+1 more|arXiv (Cornell University)|Aug 8, 2012

Formal Methods in Verification被引用 7

一句话总结

本文通过应用三种图操作——连接、顶点粘贴和单顶点添加，提出了在特定图类上求解公平游戏的多项式时间算法。证明了若某一类公平游戏可多项式时间求解，则经由这些操作得到的类同样可多项式时间求解，从而将可解性扩展至完全图的定向图、完全二分图的定向图、块图以及块-仙子图。

ABSTRACT

Parity games are games that are played on directed graphs whose vertices are labeled by natural numbers, called priorities. The players push a token along the edges of the digraph. The winner is determined by the parity of the greatest priority occurring infinitely often in this infinite play. A motivation for studying parity games comes from the area of formal verification of systems by model checking. Deciding the winner in a parity game is polynomial time equivalent to the model checking problem of the modal mu-calculus. Another strong motivation lies in the fact that the exact complexity of solving parity games is a long-standing open problem, the currently best known algorithm being subexponential. It is known that the problem is in the complexity classes UP and coUP. In this paper we identify restricted classes of digraphs where the problem is solvable in polynomial time, following an approach from structural graph theory. We consider three standard graph operations: the join of two graphs, repeated pasting along vertices, and the addition of a vertex. Given a class C of digraphs on which we can solve parity games in polynomial time, we show that the same holds for the class obtained from C by applying once any of these three operations to its elements. These results provide, in particular, polynomial time algorithms for parity games whose underlying graph is an orientation of a complete graph, a complete bipartite graph, a block graph, or a block-cactus graph. These are classes where the problem was not known to be efficiently solvable. Previous results concerning restricted classes of parity games which are solvable in polynomial time include classes of bounded tree-width, bounded DAG-width, and bounded clique-width. We also prove that recognising the winning regions of a parity game is not easier than computing them from scratch.

研究动机与目标

• 识别公平游戏可在多项式时间内求解的结构图类。
• 分析标准图操作是否保持公平游戏的多项式时间可解性。
• 将已知的可 tractable 类扩展至有界树宽和团宽之外，包括块图、块-仙子图以及完全图的定向图。
• 证明识别获胜区域并不比从头开始计算获胜区域更容易。
• 建立添加单个顶点或沿顶点粘贴游戏时，若基础类可多项式时间求解，则该性质仍保持不变。

提出的方法

• 定义三种图操作：两个游戏的连接、沿顶点的重复粘贴，以及添加单个顶点。
• 使用吸引子集和位置策略分解游戏并递归求解子游戏。
• 利用基类 C 的遗传性质，确保操作后子游戏仍属于 C。
• 采用动态规划与递归分解：求解子游戏，使用吸引子集合并获胜区域。
• 通过递推式 T(n) ≤ n² + t·nᶜ + T(n−1) 限制运行时间，得到粘贴操作的复杂度为 O(n^{1+max{2,c}})，顶点添加的复杂度为 O(n^{c+1})。
• 基于顶点策略，利用引理 3.2 和引理 3.4 证明区域合并与递归分解的合理性。

实验结果

研究问题

• RQ1 能否通过标准图操作扩展多项式时间可解的公平游戏类？
• RQ2 两个多项式时间可解类的连接是否仍为多项式时间可解？
• RQ3 沿共享顶点粘贴多个游戏是否保持多项式时间可解性？
• RQ4 向多项式时间可解类添加单个顶点是否会得到另一个多项式时间可解类？
• RQ5 在这些类中，识别获胜区域是否与从头开始计算获胜区域一样困难？

主要发现

• 完全图定向图上的公平游戏类可在多项式时间内求解。
• 完全二分图定向图上的公平游戏类可在多项式时间内求解。
• 块图和块-仙子图上的公平游戏类可在多项式时间内求解。
• 两个多项式时间可解类的连接仍可多项式时间求解，运行时间增长被多项式因子所限制。
• 沿顶点粘贴游戏保持了多项式时间可解性，运行时间增长为 O(n^{1+max{2,c}})。
• 向多项式时间可解类添加单个顶点后，所得类可在 O(n^{c+1}) 时间内求解，其中 c ≥ 2。