Packing circles in a square: a theoretical comparison of various convexification techniques
نویسنده
چکیده
We consider the problem of packing congruent circles with the maximum radius in a unit square. As a mathematical program, this problem is a notoriously difficult nonconvex quadratically constrained optimization problem which possesses a large number of local optima. We study several convexification techniques for the circle packing problem, including polyhedral and semi-definite relaxations and assess their strength theoretically. As we demonstrate both theoretically and numerically, when embedded in branch-and-cut based global solvers, the current state-of-the-art bounding techniques are only effective for small-size circle packing problems.
منابع مشابه
A better packing of ten equal circles in a square
Let S be a square of side s in the Euclidean plane. A pucking of circles in S is nothing else but a finite family of circular disks included in S whose interiors are pairwise disjoints. A natural problem related with such packings is the description of the densest ones; in particular, what is the greatest value of the common radius r of n circles forming a packing of S? Clearly, the centres of ...
متن کاملPacking Equal Circles in a Square: I. Solution Properties
In this paper some properties of optimal solutions for the problem of packing n equal circles into the unit square will be derived. In particular, properties, which must be satissed by at least one optimal solution of the problem and stating the intuitive fact that as many circles as possible should touch the boundary of the unit square, will be introduced.
متن کاملInvestigating Circles in a Square Packing Problems as a Realistic Benchmark for Continuous Metaheuristic Optimization Algorithms
In recent years, there has been a growing interest in the development of experimental methodology in metaheuristics. In the field, researchers continue to develop and propose new algorithms at a rapid rate. While theoretical analysis has made progress and continues to develop, it is often the case that algorithms are evaluated and analyzed via empirical techniques. A vital component of experime...
متن کاملSplit Packing: An Algorithm for Packing Circles with up to Critical Density
In the classic circle packing problem, one asks whether a given set of circles can be packed into the unit square. This problem is known to be NP-hard. In this thesis, we present a new sufficient condition using only the circles’ combined area: It is possible to pack any circle instance with a combined area of up to ≈53.90% of the square’s area. This area condition is tight, in the sense that f...
متن کاملStatic Symmetry Breaking in Circle Packing
We present new Static Symmetry-Breaking Inequalities (SSBI) [11,6] for the problem of packing equal circles in a square [9]. The new SSBIs provide a marked computational improvement with respect to past work [1], though not yet at the level where a purely Mathematical Programming (MP) based spatial Branch-and-Bound (sBB) can be competitive with a Branch-and-Bound (BB) “boosted” by combinatorial...
متن کامل