DMVP: Foremost Waypoint Coverage of Time-Varying Graphs

Abstract

We consider the Dynamic Map Visitation Problem (DMVP), in which a team of agents must visit a collection of critical locations as quickly as possible, in an environment that may change rapidly and unpredictably during the agents' navigation. We apply recent formulations of time-varying graphs (TVGs) to DMVP, shedding new light on the computational hierarchy R ⊃ B ⊃ P of TVG classes by analyzing them in the context of graph navigation. We provide hardness results for all three classes, and for several restricted topologies, we show a separation between the classes by showing severe inapproximability in R, limited approximability in B, and tractability in P. We also give topologies in which DMVP in R is fixed parameter tractable, which may serve as a first step toward fully characterizing the features that make DMVP

An Introduction to Temporal Graphs: An Algorithmic Perspective*

“…The 3/2 factor follows because the remaining n/2 edges that are added during the patching process cost at most n, which, in turn, is another lower bound to the cost of the optimum TSP tour. This was one of the first algorithms known for ATSP (1,2). Other approaches have improved the factor to the best currently known 5/4 [11].…”

“…In particular, there is a cost function c : A → {1, 2} assigning a cost to every time-edge of the temporal graph (see Figure 10 for an illustration). This is called the Temporal Traveling Salesman Problem with Costs One and Two and abbreviated TTSP (1,2). Now observe that the famous (static) ATSP(1,2) problem is a special case of TTSP (1,2) when the lifetime of the temporal graph D = (V , A) is restricted to n and c(e, t) = c(e, t ) for all edges e and times t, t .…”

“…This is called the Temporal Traveling Salesman Problem with Costs One and Two and abbreviated TTSP (1,2). Now observe that the famous (static) ATSP(1,2) problem is a special case of TTSP (1,2) when the lifetime of the temporal graph D = (V , A) is restricted to n and c(e, t) = c(e, t ) for all edges e and times t, t . This immediately implies that TTSP(1,2) is also APX-hard [66] and cannot be approximated within any factor less than 207/206 [36] and the same 264 MICHAIL Figure 10 An instance of TTSP(1,2) consisting of a complete temporal graph D = (V , A), where V = {u 1 , u 2 , u 3 , u 4 }, and a cost function c : A → {1, 2}, which is presented by the corresponding costs on the edges.…”

“…Other authors [73,25] have assumed that an edge might be available for a whole time-interval [t 1 , t 2 ] or several such intervals, not just for discrete moments, or that it has time-dependent travel-times [39]. Finally, [1] studied the Dynamic Map Visitation problem, in which a team of agents, operating in a dynamic environment, must visit a collection of critical locations as quickly as possible.…”

“…Though static graphs 1 have been extensively studied, for their temporal generalization we are still far from having a concrete set of structural and algorithmic principles. Additionally, it is not yet clear how the complexity of combinatorial optimization problems is affected by introducing to them a notion of time.…”