A Faster Parameterized Algorithm for Temporal Matching

Abstract

A temporal graph is a sequence of graphs (called layers) over the same vertex set -- describing a graph topology which is subject to discrete changes over time. A $Δ$-temporal matching $M$ is a set of time edges $(e,t)$ (an edge $e$ paired up with a point in time $t$) such that for all distinct time edges $(e,t),(e',t') \in M$ we have that $e$ and $e'$ do not share an endpoint, or the time-labels $t$ and $t'$ are at least $Δ$ time units apart. Mertzios et al. [STACS '20] provided a $2^{O(Δν)}\cdot |{\mathcal G}|^{O(1)}$-time algorithm to compute the maximum size of a $Δ$-temporal matching in a temporal graph $\mathcal G$, where $|\mathcal G|$ denotes the size of $\mathcal G$, and $ν$ is the $Δ$-vertex cover number of $\mathcal G$. The $Δ$-vertex cover number is the minimum number $ν$ such that the classical vertex cover number of the union of any $Δ$ consecutive layers of the temporal graph is upper-bounded by $ν$. We show an improved algorithm to compute a $Δ$-temporal matching of maximum size with a running time of $Δ^{O(ν)}\cdot |\mathcal G|$ and hence provide an exponential speedup in terms of $Δ$.