Online covering salesman problem

Given a graph , the generalized covering salesman problem (CSP) is to find a shortest tour in G such that each vertex is either on the tour or within a predetermined distance L to an arbitrary vertex on the tour, where ,. In this paper, we propose the online CSP, where the salesman will encounter at most k blocked edges during the traversal. The edge blockages are real-time, meaning that the salesman knows about a blocked edge when it occurs. We present a lower bound and a CoverTreeTraversal algorithm for online CSP which is proved to be -competitive, where , is the approximation ratio for Steiner tree problem and is the maximal number of locations that a customer can be served. When , our algorithm is near optimal. The problem is also extended to the version with service cost, and similar results are derived.