On temporally connected graphs of small cost

Akrida, Eleni C.; G{\c{a}}sieniec, Leszek; Mertzios, George B.; Spirakis, Paul G.
We study the design and verification of temporal graphs that are temporally connected. We mainly consider undirected graphs of {\$}n{\$} vertices and follow the model of $\backslash$cite{\{}kempe{\}}, 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{\$}. 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 can $\backslash$emph{\{}freely choose{\}} availability instances for all edges and aims for temporal connectivity with a very small $\backslash$emph{\{}cost{\}}; the cost here 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. Next, we consider the case in which a designer could only choose among a pre-specified set of availability instances. She comes to us and says "with those availability instances I deliver a temporally connected graph." Here, we first have to verify the correctness of the design. Then our aim is to decrease the cost by removing some labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of labels that are redundant is APX-hard, i.e., there is no PTAS unless {\$}P=NP{\$}. On the other hand, a temporal design may be "minimal" i.e. all its labels may be needed for temporal connectivity. We partially characterise minimal temporal designs.
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Type of Publication:
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
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