The Temporal Explorer Who Returns to the Base

Abstract

In this paper we study the problem of exploring a temporal graph (i.e. a graph that changes over time), in the fundamental case where the underlying static graph is a star on n vertices. The aim of the exploration problem in a temporal star is to find a temporal walk which starts at the center of the star, visits all leaves, and eventually returns back to the center. We present here a systematic study of the computational complexity of this problem, depending on the number k of time-labels that every edge is allowed to have; that is, on the number k of time points where each edge can be present in the graph. To do so, we distinguish between the decision version $\textsc {StarExp}(k)$, asking whether a complete exploration of the instance exists, and the maximization version $\textsc {MaxStarExp}(k)$ of the problem, asking for an exploration schedule of the greatest possible number of edges in the star. We fully characterize $\textsc {MaxStarExp}(k)$ and show a dichotomy in terms of its complexity: on one hand, we show that for both $k=2$ and $k=3$, it can be efficiently solved in $O(n\log n)$ time; on the other hand, we show that it is APX-complete, for every $k\ge 4$ (does not admit a PTAS, unless P = NP, but admits a polynomial-time 1.582-approximation algorithm). We also partially characterize $\textsc {StarExp}(k)$ in terms of complexity: we show that it can be efficiently solved in $O(n\log n)$ time for $k \in \{2,3\}$ (as a corollary of the solution to $\textsc {MaxStarExp}(k)$, for $k\in \{2,3\}$), but is NP-complete, for every $k\ge 6$.

Partially supported by the NeST initiative of the School of EEE and CS at the University of Liverpool and by the EPSRC Grants EP/P020372/1 and EP/P02002X/1.