Optimal covering of rectangular grid graphs with tours of constrained length

TL;DR

This paper presents a linear-time algorithm for optimally covering rectangular grid graphs with tours starting and ending at a base station, constrained by length, optimizing either the number of tours or their total length.

Contribution

It introduces a novel linear-time algorithm for optimal covering of grid graphs with length-constrained tours, addressing two different optimization objectives.

Findings

Algorithm computes optimal solutions in linear time.

Optimal covering solutions are achieved for both objectives.

The method efficiently handles large grid graphs.

Abstract

Given a rectangular grid graph with a special vertex at a corner called base station, we study the problem of covering the vertices of the entire graph with tours that start and end at the base station and whose lengths do not exceed a given threshold, while minimizing a quality measure. We consider two objective functions: minimizing the number of tours and minimizing the sum of their lengths. We present an algorithm that computes the optimal solution for both objectives in linear time with respect to the grid size.