16 - Travelling Salesman

Summary

Introduction

In this chapter, we visit a classical combinatorial problem, the travelling salesman problem (TSP). In the offline case, TSP is formulated over an undirected graph, where each edge has a weight, and the objective is to minimize the total edge weight of the tour that starts and ends at the same vertex and visits each vertex of the graph at least once. TSP in the offline case is a very rich problem and has been an object of intense study. In the online setting, TSP can be posed in multiple ways, and we study two of the most prominent versions in this chapter.

The first version we consider involves sites to be visited that belong to a metric space. Consider a walker that can walk at most unit speed. Starting at a fixed site, sites (locations) to be visited in the future belonging to a metric space are revealed sequentially, while the walker is travelling. The goal of the walker is to visit all sites and return to the starting site in the minimum time possible while ensuring that a site is visited only after it has been revealed. This version captures some of the online counterparts of the usual offline TSP applications. For this version, we first show that the competitive ratio of any online algorithm is at least 2, and then present a simple algorithm that achieves the lower bound.

The second version we consider is an exploration problem over an unknown graph. Assume that a walker is at a particular (starting) vertex of an unknown undirected edge weighted graph G. The walker's objective is to visit all the vertices of G and return to the starting vertex over the shortest path. The online restriction is that at any time, only the neighbours of all the visited vertices so far and the associated edge weights are revealed. Thus, each time a walker reaches a new vertex, it has to decide which vertex to visit next, given the partial graph information. This unknown graph exploration version is seemingly more difficult than the site exploration version, and we present the best-known online algorithm with the competitive ratio of at most 16.