Robust optimization for routing problems on trees

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 ℓ :E→ℝ_+ , a special vertex r∈ V , penalties π :V→ℝ_+ and the goal is to find a closed tour K such that r∈ V(K) and such that the cost ℓ (K)+π (V∖ 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 ξ . The cost f(K,ξ ) of a tour K in scenario ξ is the cost resulting from “shortcutting” K , i.e. from restricting K to the nodes which are reachable from r in scenario ξ , i.e. in the graph T ∖ξ =(V,E∖ξ ) . 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.