The Covering Path Problem on a Grid

TL;DR

This paper studies the covering path problem on grids, proposing approximation algorithms that balance path length and stop count, with applications in transportation such as school bus routing.

Contribution

It introduces the CPPG, a bi-objective optimization problem on grid graphs, and develops approximation methods leveraging geometric properties and problem transformations.

Findings

Provides a lower bound for the bi-objective problem.

Develops constant-factor approximation algorithms.

Applies methods to transportation problems like school bus routing.

Abstract

This paper introduces the covering path problem on a grid (CPPG) which finds the cost-minimizing path connecting a subset of points in a grid such that each point that needs to be covered is within a predetermined distance of a point from the chosen subset. We leverage the geometric properties of the grid graph which captures the road network structure in many transportation problems, including our motivating setting of school bus routing. As defined in this paper, the CPPG is a bi-objective optimization problem comprised of one cost term related to path length and one cost term related to stop count. We develop a trade-off constraint which quantifies the trade-off between path length and stop count and provides a lower bound for the bi-objective optimization problem. We introduce simple construction techniques to provide feasible paths that match the lower bound within a constant factor. Importantly, this solution approach uses transformations of the general CPPG to either a discrete CPPG or continuous CPPG based on the value of the coverage radius. For both the discrete and continuous versions, we provide fast constant-factor approximations, thus solving the general CPPG.