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

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.

💡 Research Summary

The paper addresses the NP‑hard optimal circle‑coverage problem in arbitrary convex polygons, where a fixed number of equal‑radius circles must be placed to maximize the covered area while allowing overlaps. Existing methods either focus on regular domains or suffer from poor initialization, uncontrolled boundary effects, and excessive overlap when applied to irregular shapes. To overcome these drawbacks, the authors propose the Improved Quasi‑Physical Dynamic (IQPD) algorithm, built on three novel components. First, a structure‑preserving initialization maps a hexagonal close‑packing lattice onto the target polygon via scaling and affine transformation, guaranteeing a dense, geometrically coherent starting configuration. Second, a virtual force field combines repulsive forces between circles, a friction term to damp motion, and an iterative radius‑expansion scheme that gradually enlarges circle radii while monitoring overlap, thereby directly increasing coverage rather than merely repositioning fixed‑size circles. Third, a boundary‑surrounding strategy uses normal and tangential gradient information to pull any circles that have drifted outside back inside the polygon smoothly, reducing boundary violations and eliminating “overflow” circles. Experimental evaluation on seven diverse convex polygons and three circle‑count settings compares IQPD against four state‑of‑the‑art approaches (traditional quasi‑physical, VOROPACK‑D, VGSOK, and a generic metaheuristic). Seven metrics—including total coverage ratio, average overlap, boundary violation rate, runtime, iteration count, memory usage, and final objective value—show that IQPD consistently achieves higher coverage (up to 5–7 percentage points improvement), lower overlap and boundary errors, and faster convergence (≈20–35 % reduction in time). The analysis confirms that the high‑density initialization and adaptive radius growth are the primary drivers of performance, while the gradient‑based boundary correction stabilizes the solution in complex shapes. Limitations are acknowledged: the current formulation assumes convexity and equal radii, and extending to non‑convex domains or heterogeneous circles will require more sophisticated gradient handling and multi‑scale initialization. Future work is outlined to address these extensions, explore GPU‑accelerated parallel simulations, and apply the method to real‑world scenarios such as irrigation layout, wireless sensor deployment, and emergency‑response resource allocation.