Computing maximum matchings in temporal graphs

Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G, a temporal graph is represented by assigning a set of integer time-labels to every edge e of G, indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem MAXIMUM MATCHING, taking into account the dynamic nature of temporal graphs. In our problem, MAXIMUM TEMPORAL MATCHING, we are looking for the largest possible number of time-labeled edges (simply time-edges) (e, t) such that no vertex is matched more than once within any time window of A consecutive time slots, where A is an element of N is given. We prove strong computational hardness results for MAXIMUM TEMPORAL MATCHING, even for elementary cases, as well as fixed-parameter algorithms with respect to natural parameters and polynomial-time approximation algorithms. (c) 2023 Elsevier Inc. All rights reserved.