The Traveling Salesman Problem for Systems with Dynamic Constraints

Abstract

The Traveling Salesman Problem (TSP) is a foundational problem in the fields of theoretical computer science and optimization in which an agent is tasked with visiting a set of 𝑛 target locations (in any order) in the shortest amount of time, either on a graph or in a space. As this problem is well-known to be NP-hard, it is usually solved using heuristics or approximation algorithms. An important variant of the TSP is the Dynamic TSP (DTSP), in which the targets exist in a space in which the agent’s trajectory must satisfy dynamic constraints (for instance, limited ability to accelerate). The DTSP arises naturally in many robotic motion planning problems, particularly in exploration, surveillance and reconnaissance, and is generally not amenable to the standard TSP approximation algorithms. An interesting and important question, known as the Dynamic Stochastic TSP (DSTSP), asks: if the target points are distributed randomly, how does the length of the shortest tour (either in expectation or with high probability) grow with the number 𝑛 of targets? This problem has been studied for a variety of common vehicle models, as well as certain broader classes of dynamic control systems.

In this thesis, we present a novel proof that extends known DSTSP order-of-growth results to a wider variety of dynamic systems, in particular to manifold workspaces, as well as two novel algorithms which achieve a constant-factor approximation of the optimal tour with high probability. These new proofs and algorithms furthermore allow us to study not only the order-of-growth of the tour length but also, for the important subset of ‘symmetric’ dynamics, to give explicit constant factors and to tightly characterize the relationship between the dynamics, the target point distribution, and the optimal tour length. Finally, we extend these results to the non-stochastic adversarial case, in which the target points are chosen to maximize the length of the optimal tour.