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, asking whether a complete exploration of the instance exists, and the maximization version of the problem, asking for an exploration schedule of the greatest possible number of edges in the star. We fully characterize and show a dichotomy in terms of its complexity: on one hand, we show that for both and, it can be efficiently solved in time; on the other hand, we show that it is APX-complete, for every (does not admit a PTAS, unless P NP, but admits a polynomial-time 1.582-approximation algorithm). We also partially characterize in terms of complexity: we show that it can be efficiently solved in time for (as a corollary of the solution to, for), but is NP-complete, for every.