A Faster Parameterized Algorithm for Temporal Matching

• Main Author: Zschoche, Philipp
• Format: Journal Article
• Language: English
• Published: 20.10.2020
• Subjects: Computer Science - Data Structures and Algorithms; Computer Science - Discrete Mathematics
• DOI: 10.48550/arxiv.2010.10408

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$\Delta$ .