Parameterised algorithms for temporally satisfying reconfiguration problems

TL;DR

This paper introduces fixed-parameter tractable algorithms for reconfiguring vertex-selection solutions in temporal graphs, enabling efficient reconfiguration checks and adapting approximation results from static problems.

Contribution

It presents novel fixed-parameter algorithms for temporal reconfiguration problems based on enumeration time, neighborhood diversity, lifetime, and treewidth.

Findings

Efficient algorithms for reconfiguration checks in temporal graphs.

Adaptation of static problem approximation results to temporal settings.

Fixed-parameter tractability results based on graph parameters.

Abstract

Given a static vertex-selection problem (e.g. independent set, dominating set) on a graph, we can define a corresponding temporally satisfying reconfiguration problem on a temporal graph which asks for a sequence of solutions to the vertex-selection problem at each time such that we can reconfigure from one solution to the next. We can think of each solution in the sequence as a set of vertices with tokens placed on them; our reconfiguration model allows us to slide tokens along active edges of a temporal graph at each time-step. We show that it is possible to efficiently check whether one solution can be reconfigured to another, and show that approximation results on the static vertex-selection problem can be adapted with a lifetime factor to the reconfiguration version. Our main contributions are fixed-parameter tractable algorithms with respect to: enumeration time of the related static problem; the combination of temporal neighbourhood diversity and lifetime of the input temporal graph; and the combination of lifetime and treewidth of the footprint graph.