Temporal interval cliques and independent sets

Danny Hermelin, Yuval Itzhaki, Hendrik Molter, Rolf Niedermeier

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

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.

Original languageEnglish
Article number113885
JournalTheoretical Computer Science
Volume961
DOIs
StatePublished - 15 Jun 2023

Keywords

  • Algorithms and complexity
  • Interval graphs
  • Order preservation
  • Temporal graphs
  • Vertex orderings

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science

Fingerprint

Dive into the research topics of 'Temporal interval cliques and independent sets'. Together they form a unique fingerprint.

Cite this