Distance to Transitivity: New Parameters for Taming Reachability in Temporal Graphs

TL;DR

This paper introduces new parameters measuring how close a temporal graph is to being transitive, and demonstrates their impact on the complexity of reachability and connectivity problems in temporal graphs.

Contribution

It defines novel parameters for quantifying non-transitivity in temporal graphs and analyzes their influence on the tractability of reachability-related problems.

Findings

Parameters enable fixed-parameter tractability results.

Polynomial kernels are obtained for certain problems.

Results hold without restrictions on the underlying graph or lifetime.

Abstract

A temporal graph is a graph whose edges only appear at certain points in time. Reachability in these graphs is defined in terms of paths that traverse the edges in chronological order (temporal paths). This form of reachability is neither symmetric nor transitive, the latter having important consequences on the computational complexity of even basic questions, such as computing temporal connected components. In this paper, we introduce several parameters that capture how far a temporal graph $\mathcal{G}$ is from being transitive, namely, \emph{vertex-deletion distance to transitivity} and \emph{arc-modification distance to transitivity}, both being applied to the reachability graph of $\mathcal{G}$. We illustrate the impact of these parameters on the temporal connected component problem, obtaining several tractability results in terms of fixed-parameter tractability and polynomial kernels. Significantly, these results are obtained without restrictions of the underlying graph, the snapshots, or the lifetime of the input graph. As such, our results isolate the impact of non-transitivity and confirm the key role that it plays in the hardness of temporal graph problems.