On Salesmen, Repairmen, Spiders, and Other Traveling Agents

The Traveling Salesman Problem (TSP) is a classical problem in discrete optimization. Its paradigmatic character makes it one of the most studied in computer science and operations research and one for which an impressive amount of algorithms (in particular heuristics and approximation algorithms) have been proposed. While in the general case the problem is known not to allow any constant ratio approximation algorithm and in the metric case no better algorithm than Christoodes' algorithm is known, which guarantees an approximation ratio of 3/2, recently an important breakthrough by Arora has led to the deenition of a new polynomial approximation scheme for the Euclidean case. A growing attention has also recently been posed on the approximation of other paradigmatic routing problems such as the Travelling Repairman Problem (TRP). The altruistic Travelling Repairman seeks to minimimize the average time incurred by the customers to be served rather than to minimize its working time like the egoistic Travelling Salesman does. The new approximation scheme for the Travelling Salesman is also at the basis of a new approximation scheme for the Travelling Repairman problem in the euclidean space. New interesting constant approximation algorithms have recently been presented also for the Travelling Repairman on general metric spaces. Interesting applications of this line of research can be found in the problem of routing agents over the web. In fact the problem of programming a \spider" for eeciently searching and reporting information is a clear example of potential applications of algorithms for the above mentioned problems. These problems are very close in spirit to the problem of searching an object in a known graph introduced by Koutsoupias, Papadimitriou and Yannakakis 13]. In this paper, motivated by web searching applications, we summarize the most important recent results concerning the approximate solution of the TRP and the TSP and their application and extension to web searching problems.