A graph vertex-subset problem defines which subsets of the vertices of an input graph are feasible solutions. We view a feasible solution as a set of tokens placed on the vertices of the graph. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions of size , whether it is possible to transform one into the other by a sequence of token slides (along edges of the graph) or token jumps (between arbitrary vertices of the graph) such that each intermediate set remains a feasible solution of size . Many algorithmic questions present themselves in the form of reconfiguration problems: Given the description of an initial system state and the description of a target state, is it possible to transform the system from its initial state into the target one while preserving certain properties of the system in the process? Such questions have received a substantial amount of attention under the so-called combinatorial reconfiguration framework. We consider reconfiguration variants of three fundamental underlying graph vertex-subset problems, namely , , and . We survey both older and more recent work on the parameterized complexity of all three problems when parameterized by the number of tokens . The emphasis will be on positive results and the most common techniques for the design of fixed-parameter tractable algorithms.

中文翻译：

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重构问题参数化复杂度研究


图顶点子集问题定义输入图的顶点的哪些子集是可行解。我们将可行的解决方案视为放置在图的顶点上的一组标记。顶点子集问题的重新配置变体询问，给定两个大小为 的可行解，是否可以通过一系列令牌滑动（沿着图的边缘）或令牌跳跃（在图的任意顶点之间）将一个解转换为另一个解。图）使得每个中间集仍然是大小为 的可行解。许多算法问题以重构问题的形式出现：给定初始系统状态的描述和目标状态的描述，是否有可能将系统从初始状态转换为目标状态，同时保留系统的某些属性在此过程中？这些问题在所谓的组合重构框架下受到了大量关注。我们考虑三个基本的底层图顶点子集问题的重新配置变体，即 、 和 。我们调查了有关所有三个问题（当按标记数量进行参数化时）的参数化复杂性的较旧和较新的工作。重点将放在固定参数易处理算法设计的积极结果和最常用技术上。