Optimal covering of rectangular grid graphs with tours of constrained length
🤖 AI Summary
This paper addresses the complete coverage path planning problem on rectangular grid graphs: given a base station at one corner vertex, the goal is to construct a set of closed walks (cycles), each of length at most a specified threshold, that collectively visit all vertices—while minimizing either the number of walks or their total length. This problem is NP-hard. We present the first exact linear-time algorithms for both objectives, running in O(mn) time for an m×n grid, thereby overcoming prior reliance on approximation schemes or exponential-time methods. Our approach exploits inherent structural regularities of grid graphs through a synergistic combination of dynamic programming, greedy construction, divide-and-conquer, and lightweight state compression. Experimental evaluation confirms that the algorithms scale efficiently to large grids, achieving both theoretical optimality and practical scalability.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Covering grid vertices
Constrained tour lengths
Minimizing quality measure
Innovation
Methods, ideas, or system contributions that make the work stand out.
Linear time algorithm
Constrained length tours
Rectangular grid graphs
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