Routing on Sparse Graphs with Non-metric Costs for the Prize-collecting Travelling Salesperson Problem

In many real-world routing problems, decision makers must optimise over sparse graphs such as transportation networks with non-metric costs on the edges that do not obey the triangle inequality. Motivated by finding a sufficiently long running route in a city that minimises the air pollution exposure of the runner, we study the Prize-collecting Travelling Salesperson Problem (Pc-TSP) on sparse graphs with non-metric costs. Given an undirected graph with a cost function on the edges and a prize function on the vertices, the goal of Pc-TSP is to find a tour rooted at the origin that minimises the total cost such that the total prize is at least some quota. First, we introduce heuristics designed for sparse graphs with non-metric cost functions where previous work dealt with either a complete graph or a metric cost function. Next, we develop a branch & cut algorithm that employs a new cut we call the disjoint-paths cost-cover (DPCC) cut. Empirical experiments on two datasets show that our heuristics can produce a feasible solution with less cost than a state-of-the-art heuristic from the literature. On datasets with non-metric cost functions, DPCC is found to solve more instances to optimality than the baseline cutting algorithm we compare against.