A Faster Parameterized Algorithm for Temporal Matching

• Published in: arXiv.org
• Main Author: Zschoche, Philipp
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 23.04.2021
• Subjects: Algorithms; Apexes; Graph matching; Graph theory; Topology
• ISSN: 2331-8422

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 \((e,t),(e',t') \in M\) we have that \(e\) and \(e'\) do not share an endpoint, or the time-labels \(t\) and \(t'\) are at least \(\Delta\) time units apart. Mertzios et al. [STACS '20] provided a \(2^{O(\Delta\nu)}\cdot |{\mathcal G}|^{O(1)}\)-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 \(\Delta\).