Robust optimization for routing problems on trees

Abstract

The prize-collecting travelling salesman problem (pc-tsp) is a known variant of the classical travelling salesman problem. In the pc-tsp, we are given a weighted graph $G=(V,E)$ with edge weights $\ell :E\rightarrow {\mathbb {R}}_+$, a special vertex $r\in V$, penalties $\pi :V\rightarrow {\mathbb {R}}_+$ and the goal is to find a closed tour K such that $r\in V(K)$ and such that the cost $\ell (K)+\pi (V{\setminus } V(K))$, which is the sum of the weights of the edges in the tour and the cost of the vertices not spanned by K, is minimized. In this paper, we study the pc-tsp from a viewpoint of robust optimization. In our setting, an operator must find a closed tour in a (known) edge-weighted tree $T=(V,E)$ starting and ending in the depot r while some of the edges may be blocked by “avalanches” defining the scenario $\xi$. The cost $f(K,\xi )$ of a tour K in scenario $\xi$ is the cost resulting from “shortcutting” K, i.e. from restricting K to the nodes which are reachable from r in scenario $\xi$, i.e. in the graph $T {\setminus } \xi =(V,E{\setminus }\xi )$. In the absolute robust version of the problem one searches for a tour which minimizes the worst-case cost over all scenarios, while in the deviation robust counterpart, the regret, that is, the deviation from an optimum solution for a particular scenario, is sought to be minimized. We show that both versions of the problem can be solved in polynomial time on trees.