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 ∆-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 ∆ time units apart. Mertzios et al. [STACS '20] provided a 2 O(∆ν) • |G| O(1) -time algorithm to compute the maximum size of ∆-temporal matching in a temporal graph G, where |G| denotes the size of G, and ν is the ∆-vertex cover number of G. The ∆-vertex cover number is the minimum number of vertices which are needed to hit (or cover) all edges in any ∆ consecutive layers of the temporal graph. We show an improved algorithm to compute a ∆-temporal matching of maximum size with a running time of ∆ O(ν) • |G| and hence provide an exponential speedup in terms of ∆.

Parameterized Algorithms for Diverse Multistage Problems

“…Herein, diversity is a natural goal. From a motivational point of view, an interesting direction for future research is to combine the diverse multistage setting with time windows, known from other temporal domains [1,12,30,31,41]. Here, a solution to the i-th instance should be sufficiently different from the δ previous solutions in the sequence; our work covers the case δ = 1.…”