Brief Announcement: On the Existence of δ-Temporal Cliques in Random Simple Temporal Graphs

Mertzios, George; Nikoletseas, Sotiris; Raptopoulos, Christofors; Spirakis, Paul

Abstract

We consider random simple temporal graphs in which every edge of the complete graph K_n appears once within the time interval [0,1] independently and uniformly at random. Our main result is a sharp threshold on the size of any maximum δ-clique (namely a clique with edges appearing at most δ apart within [0,1]) in random instances of this model, for any constant δ. In particular, using the probabilistic method, we prove that the size of a maximum δ-clique is approximately (2 log n)/(log 1/δ) with high probability (whp). What seems surprising is that, even though the random simple temporal graph contains Θ(n²) overlapping δ-windows, which (when viewed separately) correspond to different random instances of the Erdős-Rényi random graphs model, the size of the maximum δ-clique in the former model and the maximum clique size of the latter are approximately the same. Furthermore, we show that the minimum interval containing a δ-clique is δ-o(δ) whp. We use this result to show that any polynomial time algorithm for δ-Temporal Clique is unlikely to have very large probability of success.

• Presentation Conference Type: Conference Paper (Published)
• Conference Name: 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)
• Start Date: Jun 5, 2024
• End Date: Jun 7, 2024
• Acceptance Date: Mar 21, 2024
• Online Publication Date: May 31, 2024
• Publication Date: May 31, 2024
• Deposit Date: May 2, 2024
• Publicly Available Date: Jun 10, 2024
• Publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
• Volume: 292
• Pages: 27:1-27:5
• Series Title: Leibniz International Proceedings in Informatics (LIPIcs)
• Series ISSN: 1868-8969
• Book Title: 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)
• ISBN: 9783959773157
• DOI: https://doi.org/10.4230/LIPIcs.SAND.2024.27
• Public URL: https://durham-repository.worktribe.com/output/2431671