TY - JOUR

T1 - The Complexity of Optimal Design of Temporally Connected Graphs

AU - Akrida, Eleni C.

AU - Gąsieniec, Leszek

AU - Mertzios, George B.

AU - Spirakis, Paul G.

N1 - Funding Information:
The authors wish to thank the anonymous reviewers for their comments and suggestions that helped correct the initial manuscript and improve the quality of the work. We wish to give special thanks to the reviewers for the suggestion of the theoretical proof of Theorem 4(a), which now replaces the program code previously used in the proof. This work was supported in part by: (i) the School of EEE/CS and its NeST initiative at the University of Liverpool (i) the FET EU IP Project MULTIPLEX under contract No. 317532, and (i) the EPSRC Grant EP/K022660/1. A preliminary version of this paper appeared in the 13th Workshop on Approximation and Online Algorithms, WAOA 2015 [2].
Publisher Copyright:
© 2017, The Author(s).

PY - 2017/10/1

Y1 - 2017/10/1

N2 - 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. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. 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, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least nlogn labels.

AB - 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. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. 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, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least nlogn labels.

KW - APX-hard

KW - Minimal graph

KW - Network design

KW - Random input

KW - Temporal graphs

KW - Temporally connected

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U2 - 10.1007/s00224-017-9757-x

DO - 10.1007/s00224-017-9757-x

M3 - Article

AN - SCOPUS:85016772772

VL - 61

SP - 907

EP - 944

JO - Theory of Computing Systems

JF - Theory of Computing Systems

SN - 1432-4350

IS - 3

ER -