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

Authors George B. Mertzios , Sotiris Nikoletseas , Christoforos Raptopoulos , Paul G. Spirakis

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Discrete mathematics
Keywords
  • Simple random temporal graph
  • δ-temporal clique
  • probabilistic method

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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.

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George B. Mertzios, Sotiris Nikoletseas, Christoforos Raptopoulos, and Paul G. Spirakis. Brief Announcement: On the Existence of δ-Temporal Cliques in Random Simple Temporal Graphs. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 27:1-27:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SAND.2024.27

Author Details

George B. Mertzios
  • Department of Computer Science, Durham University, UK
Sotiris Nikoletseas
  • Computer Engineering and Informatics Department, University of Patras, Greece
Christoforos Raptopoulos
  • Department of Mathematics, University of Patras, Greece
Paul G. Spirakis
  • Department of Computer Science, University of Liverpool, UK

Funding

  • Mertzios, George B.: Supported by the EPSRC grant EP/P020372/1.
  • Raptopoulos, Christoforos: Supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “2nd Call for H.F.R.I. Research Projects to support Post-Doctoral Researchers” (Project Number: 704).
  • Spirakis, Paul G.: Supported by the EPSRC grant EP/P02002X/1.

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