The prize collecting traveling salesman problem

TLDR

The PCTSP models a salesman who travels between cities at pairwise costs, collects prizes at visited cities, pays penalties for unvisited ones, and seeks to minimize total cost while meeting a prize target, representing a key class of scheduling and routing problems. The paper studies structural properties of the PCTSP polytope, the convex hull of feasible solutions. The authors identify facet‑defining inequalities that can serve as cutting planes or as components of a Lagrangian relaxation in algorithms for the PCTSP. The study identifies several families of facet‑defining inequalities for the PCTSP polytope, including ones related to the ordinary TSP and knapsack polytopes.

Abstract

Abstract The following is a valid model for an important class of scheduling and routing problems. A salesman who travels between pairs of cities at a cost depending only on the pair, gets a prize in every city that he vitis and pays a penalty to every city that he fails to visit, wishes to minimize his travel costs and net penalties, while visiting enough cities to collect a prescribed amount of prize money. We call this problem the Prize Collecting Traveling Salesman Problem (PCTSP). This paper discusses structural properties of the PCTS polytope, the convex hull of solutions to the PCTSP. In particular, it identifies several families of facet defining inequalities for this polytope. Some of these inequalities are related to facets of the ordinary TS polytope, others to facets of the knapsack polytope. They can be used in algorithms for the PCTSP either as cutting planes or as ingredients of a Lagrangean optimand.