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 $\Delta$-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 $(e,t),(e&#39;,t&#39;) \in M$ we have that $e$ and $e&#39;$ do not share an endpoint, or the time-labels $t$ and $t&#39;$ are at least $\Delta$ time units apart. Mertzios et al. [STACS '20] provided a $$2^{{O(\Delta\nu)}\cdot} |{\mathcal G}|<sup>{O(1)}$$-time algorithm to compute the maximum size of a $\Delta$-temporal matching in a temporal graph $\mathcal G$, where $|\mathcal G|$ denotes the size of $\mathcal G$, and $\nu$ is the $\Delta$-vertex cover number of $\mathcal G$. The $\Delta$-vertex cover number is the minimum number $\nu$ such that the classical vertex cover number of the union of any $\Delta$ consecutive layers of the temporal graph is upper-bounded by $\nu$. We show an improved algorithm to compute a $\Delta$-temporal matching of maximum size with a running time of $\Delta^{{O(\nu)}\cdot} |\mathcal G|$ and hence provide an exponential speedup in terms of $\Delta$.