Online covering salesman problem

Abstract

Given a graph $G=(V,E,D,W)$, the generalized covering salesman problem (CSP) is to find a shortest tour in G such that each vertex $i\in D$ is either on the tour or within a predetermined distance L to an arbitrary vertex $j\in W$ on the tour, where $D\subset V$,$W\subset V$. 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 $\frac{1}{1 + (k + 2)L}k+1$ and a CoverTreeTraversal algorithm for online CSP which is proved to be $k+\alpha$-competitive, where $\alpha =0.5+\frac{(4k+2)L}{OPT}+2\gamma \rho$, $\gamma$ is the approximation ratio for Steiner tree problem and $\rho$ is the maximal number of locations that a customer can be served. When $\frac{L}{\texttt {OPT}}\rightarrow 0$, our algorithm is near optimal. The problem is also extended to the version with service cost, and similar results are derived.