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.