Online traveling salesman problems with service flexibility

The Traveling Salesman Problem (TSP) is a well-known combinatorial optimization problem. We are concerned here with online versions of this problem defined on metric spaces. One novel aspect in the paper is the introduction of a sound theoretical model to incorporate “yes-no” decisions on which requests to serve, together with an online strategy to visit the accepted requests. In order to do so, we assume that there is a penalty for not serving a request. Requests for visit of points in the metric space are revealed over time to a server, initially at a given origin, who must decide in an online fashion which requests to serve in order to minimize the time to serve all accepted requests plus the sum of the penalties associated with the rejected requests. We first look at the special case of the non-negative real line. After providing a polynomial time algorithm for the offline version of the problem, we propose and prove the optimality of a 2-competitive polynomial time online algorithm based on re-optimization approaches. We also consider the impact of advanced information (lookahead) on this optimal competitive ratio. We then consider the generalizations of these results to the case of the real line. We show that the previous algorithm can be extended to an optimal 2-competitive online algorithm. Finally we consider the case of a general metric space and propose an original c-competitive online algorithm, where c = √ 17+5 4 ≈ 2.28. We also give a polynomial-time (1.5ρ + 1)-competitive online algorithm which uses a polynomial-time ρ-approximation for the offline problem. ∗To appear in Networks. Research funded in part by ONR, grant N00014-09-1-0326 ‡Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, and Operations Research Center, MIT, Cambridge, MA, 02139; [email protected] §Operations Research Center, MIT, Cambridge, MA, 02139; [email protected]