The temporal explorer who returns to the base☆
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 labels per edge
In this section we show that, when every edge has two or three labels, a maximum size exploration in can be efficiently found in 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 labels per edge
In this section we show that, whenever , StarExp(k) is NP-complete. Furthermore, we show that MaxStarExp(k) is APX-hard for . 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 of n vertices, where the labels are assigned to the edges of at random. In particular, each edge of 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 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)
- et al.
Temporal flows in temporal networks
J. Comput. Syst. Sci.
(2019) - et al.
Ephemeral networks with random availability of links: the case of fast networks
J. Parallel Distrib. Comput.
(2016) - et al.
Temporal vertex covers and sliding time windows
J. Comput. Syst. Sci.
(2020) - et al.
Traveling salesman problems in temporal graphs
Theor. Comput. Sci.
(2016) - et al.
Optimization, approximation, and complexity classes
J. Comput. Syst. Sci.
(1991) - et al.
Computing maximal cliques in link streams
Theor. Comput. Sci.
(2016) - et al.
DMVP: foremost waypoint coverage of time-varying graphs
- et al.
The complexity of optimal design of temporally connected graphs
Theory Comput. Syst.
(2017) - et al.
On verifying and maintaining connectivity of interval temporal networks
- et al.
An O(log n/log log n)-approximation algorithm for the asymmetric traveling salesman problem
Oper. Res.
(2017)
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Dynamic traveling repair problem with an arbitrary time window
Restricted shortest path in temporal graphs
On exploring temporal graphs of small pathwidth
Computing shortest, fastest, and foremost journeys in dynamic networks
Int. J. Found. Comput. Sci.
Deterministic Algorithms in Dynamic Networks: Formal Models and Metrics
Deterministic Algorithms in Dynamic Networks: Problems, Analysis, and Algorithmic Tools
Time-varying graphs and dynamic networks
Int. J. Parallel Emerg. Distrib. Syst.
Fast convergence for consensus in dynamic networks
ACM Trans. Algorithms
Worst-case analysis of a new heuristic for the travelling salesman problem
Approximation algorithms for the job interval selection problem and related scheduling problems
Math. Oper. Res.
Flooding time of edge-Markovian evolving graphs
SIAM J. Discrete Math.
Fine-grained complexity analysis of two classic TSP variants
Another look at the shoelace TSP: the case of very old shoes
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2023, Journal of Computer and System SciencesColorful path detection in vertex-colored temporal
2023, Network ScienceCollision-Free Robot Scheduling
2024, arXivAn FPT Algorithm for Temporal Graph Untangling
2023, Leibniz International Proceedings in Informatics, LIPIcs
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A preliminary version appeared at the 11th International Conference on Algorithms and Complexity (CIAC 2019).