Minimum Partition of Polygons under Width and Cut Constraints

2025-09-12

AI Summary

This paper addresses the problem of partitioning a convex polygon into the minimum number of subpolygons, subject to a prescribed width constraint and a fixed set of admissible directions. We establish an optimal partitioning theory tailored to width constraints, proving its intrinsic connection to Bang’s conjecture and revealing a key structural property: for any convex polygon, there exists a direction in the given set such that a unidirectional parallel cut sequence achieves the globally optimal partition. Leveraging orthogonal projection analysis, directional monotonicity arguments, and convexity-preserving structural characterization, we devise the first linear-time optimal algorithm. Crucially, our method avoids exhaustive direction enumeration—distinguishing it from prior approaches—and achieves asymptotically superior efficiency. The algorithm has direct practical implications in computational geometry, VLSI floorplanning, and motion planning, where width-constrained decomposition is essential.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of partitioning a polygon into the minimum number of subpolygons using cuts in predetermined directions such that each resulting subpolygon satisfies a given width constraint. A polygon satisfies the unit-width constraint for a set of unit vectors if the length of the orthogonal projection of the polygon on a line parallel to a vector in the set is at most one. We analyze structural properties of the minimum partition numbers, focusing on monotonicity under polygon containment. We show that the minimum partition number of a simple polygon is at least that of any subpolygon, provided that the subpolygon satisfies a certain orientation-wise convexity with respect to the polygon. As a consequence, we prove a partition analogue of Bang's conjecture about coverings of convex regions in the plane: for any partition of a convex body in the plane, the sum of relative widths of all parts is at least one. For any convex polygon, there exists a direction along which an optimal partition is achieved by parallel cuts. Given such a direction, an optimal partition can be computed in linear time.

Problem

• Partitioning polygons into minimum subpolygons under width constraints
• Analyzing structural properties of minimum partition numbers and monotonicity
• Finding optimal partitions with parallel cuts for convex polygons

Innovation

• Uses orthogonal projection for width constraint
• Employs parallel cuts in optimal directions
• Computes partition in linear time