Temporal graphs are graphs whose edges are labelled with times at which they are active. Their time-sensitivity provides a useful model of real networks, but renders many problems studied on temporal graphs more computationally complex than their static counterparts. To contend with this, there has been recent work devising parameters for which temporal problems become tractable. One such parameter is vertex-interval-membership (VIM) width. Broadly, this gives a bound on the number of vertices we need to keep track of at any given time to solve many problems. Our contributions are two-fold. Firstly, we introduce a new parameter, tree-interval-membership (TIM) width, that generalises both VIM width and several existing generalisations. Secondly, we provide meta-algorithms for both VIM and TIM width which can be used to prove fixed-parameter-tractability for large families of problems, bypassing the need to give involved dynamic programming arguments for every problem. In doing this, we provide a characterisation of problems in FPT with respect to both parameters. We apply these algorithms to temporal versions of Hamiltonian path, dominating set, matching, and edge deletion to limit maximum reachability.

翻译：时间图是一种图的边被标记为活跃时间的图。其时间敏感性为实际网络提供了有效模型，但使得时间图上的许多问题在计算上比其静态对应物更复杂。为应对这一挑战，近期研究提出了若干参数，使时间问题变得易解。其中一个参数是顶点区间隶属（VIM）宽度。广义而言，该参数限定了在任意时刻需要追踪的顶点数量，从而求解许多问题。我们的贡献有两方面。首先，我们引入了一个新参数——树区间隶属（TIM）宽度，该参数推广了VIM宽度和若干现有扩展。其次，我们为VIM和TIM宽度提供了元算法，可用于证明大规模问题族的固定参数易解性，从而避免为每个问题提供复杂的动态规划论证。在此过程中，我们刻画了关于这两个参数的FPT问题特征。我们将这些算法应用于哈密顿路、支配集、匹配以及限制最大可达性的边删除等时间版本的问题。