Traveling Salesman Problems in Temporal Graphs

Abstract

In this work, we introduce the notion of time to some well-known combinatorial optimization problems. In particular, we study problems defined on temporal graphs. A temporal graph D = (V,A) may be viewed as a time-sequence G 1,G 2…,G l of static graphs over the same (static) set of nodes V. Each G t = D(t) = (V,A(t)) is called the instance of D at time t and l is called the lifetime of D. Our main focus is on analogues of traveling salesman problems in temporal graphs. A sequence of time-labeled edges (e.g. a tour) is called temporal if its labels are strictly increasing. We begin by considering the problem of exploring the nodes of a temporal graph as soon as possible. In contrast to the positive results known for the static case, we prove that, it cannot be approximated within cn, for some constant c > 0, in general temporal graphs and within (2 − e), for every constant e > 0, in the special case in which D(t) is connected for all 1 ≤ t ≤ l, both unless P = NP. We then study the temporal analogue of TSP(1,2), abbreviated TTSP(1,2), where, for all 1 ≤ t ≤ l, D(t) is a complete weighted graph with edge-costs from {1,2} and the cost of an edge may vary from instance to instance. The goal is to find a minimum cost temporal TSP tour. We give several polynomial-time approximation algorithms for TTSP(1,2). Our best approximation is (1.7 + e) for the generic TTSP(1,2) and (13/8 + e) for its interesting special case in which the lifetime of the temporal graph is restricted to n. In the way, we also introduce temporal versions of other fundamental combinatorial optimization problems, for which we obtain polynomial-time approximation algorithms and hardness results.