An Improved Quasi-Physical Dynamic Algorithm for Efficient Circular Coverage in Arbitrary Convex

TL;DR

The paper tackles the NP-hard problem of maximizing the coverage of a convex polygon by a fixed set of congruent circles. It introduces the Improved Quasi-Physical Dynamic (IQPD) algorithm, which combines a structure-preserving hexagonal initialization, a virtual-force field with friction and a radius-expansion cycle, and a boundary-encirclement strategy driven by normal and tangential gradients. Key contributions include a hexagonal-based, affine-transformed initialization that preserves interior packing, an adaptive expansion–equilibration mechanism to reduce overlap, and an augmented-Lagrangian boundary handling that repositions overflow circles along the boundary. Empirical results show IQPD significantly outperforms four state-of-the-art methods on seven metrics across diverse convex polygons, supporting its potential for practical coverage and resource-allocation tasks.

Abstract

The optimal circle coverage problem aims to find a configuration of circles that maximizes the covered area within a given region. Although theoretical optimal solutions exist for simple cases, the problem's NP-hard characteristic makes the problem computationally intractable for complex polygons with numerous circles. Prevailing methods are largely confined to regular domains, while the few algorithms designed for irregular polygons suffer from poor initialization, unmanaged boundary effects, and excessive overlap among circles, resulting in low coverage efficiency. Consequently, we propose an Improved Quasi-Physical Dynamic(IQPD) algorithm for arbitrary convex polygons. Our core contributions are threefold: (1) proposing a structure-preserving initialization strategy that maps a hexagonal close-packing of circles into the target polygon via scaling and affine transformation; (2) constructing a virtual force field incorporating friction and a radius-expansion optimization iteration model; (3) designing a boundary-surrounding strategy based on normal and tangential gradients to retrieve overflowing circles. Experimental results demonstrate that our algorithm significantly outperforms four state-of-the-art methods on seven metrics across a variety of convex polygons. This work could provide a more efficient solution for operational optimization or resource allocation in practical applications.

Figures (7)

• Figure 1: Construction of Voronoi diagrams through half-plane intersections
• Figure 2: Two Types of Largest Empty Circle
• Figure 3: A convex polygon composed of clockwise-oriented directed line segments
• Figure 4: The two types of elastic forces
• Figure 5: Regular hexagonal close packing of circles generated from the center outward.

Theorems & Definitions (18)

• Definition 1: Convex Set
• Definition 2: Convex Hull
• Lemma 1
• Lemma 2
• Lemma 3
• Definition 3: Largest Empty Circle
• Theorem 1
• Lemma 4
• Definition 4: Signed Distance Function
• Definition 5: Right Normal Vector