Online Traveling Salesman Problems with Flexibility

Abstract

The Traveling Salesman Problem (TSP) is a well-known combinatorial 
optimization problem. We are concerned here with online versions of a 
generalization of the TSP on metric spaces where the server doesn't have 
to accept all requests. Associated with each request (to visit a point in the 
metric space) is a penalty (incurred if the request is rejected). Requests 
are revealed over time to a server, initially at a given origin, who must 
decide 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. 

In a first online version of this problem (basic version), we assume that the 
server's decision to accept or reject a request can be made any time after 
its release date. In a second online version of this problem (real-time 
version), we assume that the server's decision to accept or reject a 
request must be made exactly at its release date.

After reviewing prior results on the online TSP, we first provide an optimal 
2-competitive online algorithm for the basic version of the problem in a 
general metric space, improving prior results from the literature.  We then
consider the real-time version of the problem and show that there can't be 
any finite $c$-competitive online algorithm in a general metric space.