The temporal explorer who returns to the base

https://doi.org/10.1016/j.jcss.2021.04.001Get rights and content

Abstract

We study here the problem of exploring a temporal graph when the underlying graph is a star. The aim of the exploration problem in a temporal star is finding a temporal walk which starts and finishes at the center of the star, and visits all leaves. We present a systematic study of the computational complexity of this problem, depending on the number k of time points where each edge can be present in the graph. We distinguish between the decision version StarExp(k), asking whether a complete exploration exists, and the maximization version MaxStarExp(k), asking for an exploration of the greatest possible number of edges. We fully characterize MaxStarExp(k) in terms of complexity. We also partially characterize StarExp(k), showing that it is in P for k<4, but is NP-complete, for every k>5. Finally, we partially characterize classes of “random” temporal stars which are, asymptotically almost surely, yes-instances and no-instances for StarExp(k).

Section snippets

Introduction and motivation

A temporal graph is, roughly speaking, a graph that changes over time. Several networks, both modern and traditional, including social networks, transportation networks, information and communication networks, can be modeled as temporal graphs. The common characteristic in all the above examples is that the network structure, i.e. the underlying graph topology, is subject to discrete changes over time. Temporal graphs naturally model such time-varying networks using time-labels on the edges of

Efficient algorithm for k3 labels per edge

In this section we show that, when every edge has two or three labels, a maximum size exploration in (Gs,L) can be efficiently found in O(nlogn) time. To do that, we reduce our problem to the Interval Scheduling Maximization Problem (ISMP).

We can then apply a known greedy algorithm that finds an optimal solution for ISMP; the basic idea of this algorithm is to order the set of intervals in increasing order of finish time and then “greedily” process them in one pass, selecting as large a

Hardness for k4 labels per edge

In this section we show that, whenever k6, StarExp(k) is NP-complete. Furthermore, we show that MaxStarExp(k) is APX-hard for k4. Thus, in particular, MaxStarExp(k) does not admit a Polynomial-Time Approximation Scheme (PTAS), unless P = NP. In fact, due to a known polynomial-time constant-factor approximation algorithm for JISP(k) [18], it follows that MaxStarExp(k) is also APX-complete.

k Random labels per edge

We now study the problem of star exploration in a temporal star graph on an underlying star graph Gs of n vertices, where the labels are assigned to the edges of Gs at random. In particular, each edge of Gs receives k labels independently of other edges, and each label is chosen uniformly at random and independently from a set of available labels. We will distinguish between two models: the “integer labels” model, where the labels are integer numbers chosen from the set of positive integers up

Conclusions and open problems

In this paper, we have thoroughly investigated the computational complexity landscape of the temporal star exploration problems StarExp(k) and MaxStarExp(k), depending on the maximum number k of labels allowed per edge.

We have shown that an optimal solution to the maximization problem, on instances every edge of which has two or three labels, can be efficiently found in O(nlogn) time. This immediately implies that the decision version can be also solved in the same time. We show that StarExp(k)

CRediT authorship contribution statement

This was very much a collaborative piece of work, and as such, the roles of the authors cannot be divided particularly clearly. All authors have participated in each of the following: Methodology, Investigation, Writing – Original draft preparation, Reviewing and Editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported in part by:

  • (i)

    the NeST initiative of the EEE/CS School of the University of Liverpool,

  • (ii)

    the EPSRC grant on “Algorithmic Aspects of Temporal Graphs” EP/P020372/1,

  • (iii)

    the EPSRC grant on “Algorithmic Aspects of Temporal Graphs” EP/P02002X/1, and

  • (iv)

    the EPSRC AlgoUK network grant EP/R005613/1.

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

References (49)

  • G. Ausiello et al.

    Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties

    (1999)
  • Y. Azar et al.

    Dynamic traveling repair problem with an arbitrary time window

  • S. Biswas et al.

    Restricted shortest path in temporal graphs

  • H.L. Bodlaender et al.

    On exploring temporal graphs of small pathwidth

  • B.-M. Bui-Xuan et al.

    Computing shortest, fastest, and foremost journeys in dynamic networks

    Int. J. Found. Comput. Sci.

    (2003)
  • A. Casteigts et al.

    Deterministic Algorithms in Dynamic Networks: Formal Models and Metrics

    (2013)
  • A. Casteigts et al.

    Deterministic Algorithms in Dynamic Networks: Problems, Analysis, and Algorithmic Tools

    (2013)
  • A. Casteigts et al.

    Time-varying graphs and dynamic networks

    Int. J. Parallel Emerg. Distrib. Syst.

    (2012)
  • T.H. Chan et al.

    Fast convergence for consensus in dynamic networks

    ACM Trans. Algorithms

    (2014)
  • N. Christofides

    Worst-case analysis of a new heuristic for the travelling salesman problem

    (1976)
  • J. Chuzhoy et al.

    Approximation algorithms for the job interval selection problem and related scheduling problems

    Math. Oper. Res.

    (2006)
  • A.E.F. Clementi et al.

    Flooding time of edge-Markovian evolving graphs

    SIAM J. Discrete Math.

    (2010)
  • M. de Berg et al.

    Fine-grained complexity analysis of two classic TSP variants

  • V.G. Deineko et al.

    Another look at the shoelace TSP: the case of very old shoes

  • Cited by (23)

    • Untangling temporal graphs of bounded degree

      2023, Theoretical Computer Science
    • Parameterised temporal exploration problems

      2023, Journal of Computer and System Sciences
    • An FPT Algorithm for Temporal Graph Untangling

      2023, Leibniz International Proceedings in Informatics, LIPIcs
    View all citing articles on Scopus

    A preliminary version appeared at the 11th International Conference on Algorithms and Complexity (CIAC 2019).

    View full text