Agility and Target Distribution in the Dynamic Stochastic Traveling Salesman Problem

TL;DR

This paper investigates the Dynamic Stochastic Traveling Salesman Problem, establishing bounds on tour length based on target distribution and system dynamics, with extensions to adversarial target placement.

Contribution

It provides a more precise lower bound and a matching upper bound for symmetric systems, and extends results to adversarial target distributions.

Findings

Derived a lower bound on tour length based on target distribution and system dynamics.

Established a matching upper bound for symmetric (driftless) systems.

Extended bounds to worst-case scenarios with adversarial target placement.

Abstract

An important variant of the classic Traveling Salesman Problem (TSP) is the Dynamic TSP, in which a system with dynamic constraints is tasked with visiting a set of n target locations (in any order) in the shortest amount of time. Such tasks arise naturally in many robotic motion planning problems, particularly in exploration, surveillance and reconnaissance, and classical TSP algorithms on graphs are typically inapplicable in this setting. An important question about such problems is: if the target points are random, what is the length of the tour (either in expectation or as a concentration bound) as n grows? This problem is the Dynamic Stochastic TSP (DSTSP), and has been studied both for specific important vehicle models and for general dynamic systems; however, in general only the order of growth is known. In this work, we explore the connection between the distribution from which the targets are drawn and the dynamics of the system, yielding a more precise lower bound on tour length as well as a matching upper bound for the case of symmetric (or driftless) systems. We then extend the symmetric dynamics results to the case when the points are selected by a (non-random) adversary whose goal is to maximize the length, thus showing worst-case bounds on the tour length.