A faster parameterized algorithm for temporal matching - Zschoche, Philipp (Information processing letters, 2022)
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′)∈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 2O(Δν)⋅|G|O(1)-time algorithm to compute the maximum size of a Δ-temporal matching in a temporal graph G, where |G| denotes the size of G, and ν is the Δ-vertex cover number of 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 ΔO(ν)⋅|G| and hence provide an exponential speedup in terms of Δ. •An exponential speedup for Temporal Matching compared to the algorithm from Mertzios et al. [STACS '20].•Representing Δ-windows in a tree structure.