A Faster Parameterized Algorithm for Temporal Matching - Zschoche, Philipp (2020)
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$2^{O(Δν)}· |{\mathcal G}|^{O(1)}$ -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$Δ^{O(ν)}· |\mathcal G|$and hence provide an exponential speedup in terms of$Δ$ .