On temporally connected graphs of small cost

Eleni C. Akrida, Leszek Gąsieniec, George B. Mertzios, Paul G. Spirakis

Research output: Chapter in Book/Report/Conference proceedingConference contribution

6 Scopus citations

Abstract

We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of n vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex u to vertex v is a path from u to v where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (u, v)-journey for any pair of vertices u, v, u ≠ v. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number of availability instances used. We achieve this via a simple polynomial-time procedure which derives designs of cost linear in n, and at most the optimal cost plus 2. To show this, we prove a lower bound on the cost for any undirected graph. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead given a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless P = NP. On the positive side, we show that in dense graphs with random edge availabilities, all but Θ(n) labels are redundant whp. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least n log n labels.

Original languageEnglish (US)
Title of host publicationApproximation and Online Algorithms - 13th International Workshop, WAOA 2015, Revised Selected Papers
EditorsMartin Skutella, Laura Sanità
PublisherSpringer Verlag
Pages84-96
Number of pages13
ISBN (Print)9783319286839
DOIs
StatePublished - Jan 1 2015
Event13th International Workshop on Approximation and Online Algorithms, WAOA 2015 - Patras, Greece
Duration: Sep 17 2015Sep 18 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9499
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference13th International Workshop on Approximation and Online Algorithms, WAOA 2015
CountryGreece
CityPatras
Period9/17/159/18/15

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Akrida, E. C., Gąsieniec, L., Mertzios, G. B., & Spirakis, P. G. (2015). On temporally connected graphs of small cost. In M. Skutella, & L. Sanità (Eds.), Approximation and Online Algorithms - 13th International Workshop, WAOA 2015, Revised Selected Papers (pp. 84-96). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9499). Springer Verlag. https://doi.org/10.1007/978-3-319-28684-6_8