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

Patrick O'Hara, M. S. Ramanujan, Theodoros Damoulas

TL;DR

This work introduces heuristics designed for sparse graphs with non-metric cost functions where previous work dealt with either a complete graph or a metric cost function and develops a branch&cut algorithm that employs a new cut called the disjoint-paths cost-cover (DPCC) cut.

Abstract

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.

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

TL;DR

This work introduces heuristics designed for sparse graphs with non-metric cost functions where previous work dealt with either a complete graph or a metric cost function and develops a branch&cut algorithm that employs a new cut called the disjoint-paths cost-cover (DPCC) cut.

Abstract

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.

Paper Structure

This paper contains 37 sections, 10 theorems, 9 equations, 3 figures, 16 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Problem definition
  4. Heuristics
  5. Suurballe's Heuristic
  6. Path Extension
  7. Path Collapse
  8. Suurballe's Path Extension & Collapse (SBL-PEC)
  9. Branch & Cut Algorithm
  10. Integer Programming Formulation
  11. Cost-cover Inequalities
  12. Computational experiments
  13. London Air Quality (LAQ)
  14. TSPLIB
  15. Empirical Heuristic Results
  16. ...and 22 more sections

Key Result

Lemma 0

Let $G$ be a connected, undirected graph and $c: E(G) \to \mathbb{N}_0$ be a cost function on the edges. The number of metric edges in $E(G)$ is at least $n-1$.

Figures (3)

  • Figure 1: Example of a 10km tour that minimises the air pollution exposure of a runner in London, UK. The route starts and ends at the red vertex. Yellow cells indicate high air pollution. Blue cells indicates low air pollution.
  • Figure 2: Demonstration of the SBL and path extension stages of PEC-SBL heuristic (Algorithm 1). Vertex $0$ is the root. The prize of every vertex is 1. Edges are labelled with their cost. The quota is $Q=6$.
  • Figure 3: A flow diagram of our branch & cut algorithm. The dashed square box denotes operations that are repeated within a single node of the branch & bound tree.

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 0
  • Definition 4
  • Proposition 0
  • proof
  • Lemma 0
  • proof
  • Proposition 0
  • ...and 13 more