Online traveling salesman problems with service flexibility

In this paper we consider online versions of the Traveling Salesman Problem (TSP) on metric spaces for which requests to visit points are not mandatory. Associated with each request is a penalty (if rejected). Requests are revealed over time (at their release dates) to a server who must decide which requests to accept and serve in order to minimize a linear combination of the time to serve all accepted requests and the total penalties of all rejected requests. In the basic online version of the problem, a request can be accepted any time after its release date. In the real-time online version, a request must be accepted or rejected at the time of its release date. For the basic version, we provide a best possible 2-competitive online algorithm for the problem on a general metric space. For the real-time version, we first consider special metric spaces: on the non-negative real line, we provide a best possible 2.5-competitive polynomial time online algorithm; on the real line, we prove a lower bound of 2.64 on any competitive ratios and give a 3-competitive online algorithm. We then consider the case of a general metric space and prove a Ω( lnn) lower bound on the competitive ratio of any online algorithms. Finally, among the restricted class of online algorithms with prior knowledge about the total number of requests n, we propose an asymptotically best possible O( lnn)-competitive algorithm. key words: online, competitive analysis, traveling salesman problem