Temporal interval cliques and independent sets

• Main Authors: Hermelin, D. (Author), Itzhaki, Y. (Author), Molter, H. (Author), Niedermeier, R. (Author)
• Format: Article
• Language: English
• Published: Elsevier B.V. 2023
• Subjects: Algorithms and complexity; Approximation algorithms; Computational complexity; FPT algorithms; Graph theory; Graphic methods; Independent set; Interval graph; Interval graphs; Order preservation; Parameter estimation; Temporal graphs; Temporal intervals; Time interval; Unit interval graphs; Vertex ordering; Vertex orderings
• Online Access: View Fulltext in Publisher; View in Scopus

 Temporal graphs have been recently introduced to model changes in a given network that occur throughout a fixed period of time. The TEMPORAL Δ CLIQUE problem, which generalizes the well known CLIQUE problem to temporal graphs, has been studied in the context of finding nodes of interest in dynamic networks [TCS '16]. We introduce the TEMPORAL Δ INDEPENDENT SET problem, a temporal generalization of INDEPENDENT SET. This problem is e.g. motivated in the context of finding conflict-free schedules for maximum subsets of tasks, that have certain (time-varying) constraints within a given time period. We are specifically interested in the case where each task needs to be performed in a certain time-interval on each day and two tasks are in conflict on a certain day if their time-intervals on that day overlap. This leads us to consider both problems on the restricted class of temporal unit interval graphs, i.e., temporal graphs where each layer is a unit interval graph. We present several hardness results as well as positive results. On the algorithmic side, we provide constant-factor approximation algorithms for instances of both problems where τ, the total number of time steps (layers) of the temporal graph, and Δ, a parameter that allows us to model conflict tolerance, are constants. We develop an exact FPT algorithm for TEMPORAL Δ CLIQUE with respect to parameter τ+k. Finally, we use the notion of order preservation for temporal unit interval graphs that, informally, requires the intervals of every layer to obey a common ordering. For both problems, we provide an FPT algorithm parameterized by the size of minimum vertex deletion set to order preservation. © 2023 Elsevier B.V.