# On the Constrained Least-cost Tour Problem

## TL;DR

This paper introduces the Constrained Least-cost Tour problem, proves its NP-hardness, and develops heuristics for finding low-cost, weight-feasible routes, with applications in urban pollution exposure reduction.

## Contribution

It defines the CLT problem, proves its NP-hardness, and proposes heuristics including relaxations and cycle-finding algorithms for practical solutions.

## Key findings

- Proved CLT is NP-hard even on simple paths.
- Developed heuristics using relaxations and Suurballe's algorithm.
- Demonstrated algorithms on real-world urban pollution routing.

## Abstract

We introduce the Constrained Least-cost Tour (CLT) problem: given an undirected graph with weight and cost functions on the edges, minimise the total cost of a tour rooted at a start vertex such that the total weight lies within a given range. CLT is related to the family of Travelling Salesman Problems with Profits, but differs by defining the weight function on edges instead of vertices, and by requiring the total weight to be within a range instead of being at least some quota. We prove CLT is $\mathcal{NP}$-hard, even in the simple case when the input graph is a path. We derive an informative lower bound by relaxing the integrality of edges and propose a heuristic motivated by this relaxation. For the case that requires the tour to be a simple cycle, we develop two heuristics which exploit Suurballe's algorithm to find low-cost, weight-feasible cycles. We demonstrate our algorithms by addressing a real-world problem that affects urban populations: finding routes that minimise air pollution exposure for walking, running and cycling in the city of London.