Greedy vacancy search algorithm for packing equal circles in a square

This paper considers the problem of finding the densest packings of N(N=1,2,...) equal circles in a square. We propose a physically inspired model to formulate this problem and a new heuristic algorithm to solve this problem. The approach is tested on the instances of N=1,2,...,200. Though many researchers have searched these instances using various methods, we can still find 41 better packings than the best-known ones reported in literature.

[1]  Ronald L. Graham,et al.  Improving Dense Packings of Equal Disks in a Square , 2000, Electron. J. Comb..

[2]  R. Peikert,et al.  Packing circles in a square: A review and new results , 1992 .

[3]  Mihály Csaba Markót.  Interval methods for verifying structural optimality of circle packing configurations in the unit square , 2007 .

[4]  Patric R. J. Östergård,et al.  More Optimal Packings of Equal Circles in a Square , 1999, Discret. Comput. Geom..

[5]  Patric R. J. Östergård,et al.  Packing up to 50 Equal Circles in a Square , 1997, Discret. Comput. Geom..

[6]  Tibor Csendes,et al.  A New Verified Optimization Technique for the "Packing Circles in a Unit Square" Problems , 2005, SIAM J. Optim..

[7]  Ronald L. Graham,et al.  Dense Packings of Equal Disks in an Equilateral Triangle: from 22 to 34 and Beyond , 1995, Electron. J. Comb..

[8]  Ronald L. Graham,et al.  Repeated Patterns of Dense Packings of Equal Disks in a Square , 1996, Electron. J. Comb..

[9]  Panos M. Pardalos,et al.  New results in the packing of equal circles in a square , 1995, Discret. Math..

[10]  Kenneth Falconer,et al.  Unsolved Problems In Geometry , 1991 .

[11]  J. Henry,et al.  System Modelling and Optimization: Proceedings of the 16th IFIP-TC7 Conference, Compiègne, France, July 5-9, 1993 , 1994, System Modelling and Optimization.

[12]  Tibor Csendes,et al.  Packing Equal Circles in a Square II. — New Results for up to 100 Circles Using the TAMSASS-PECS Algorithm , 2001 .

[13]  Jorge Nocedal,et al.  On the limited memory BFGS method for large scale optimization , 1989, Math. Program..

[14]  Ira M. Gessel,et al.  The Tutte polynomial of a graph, depth-first search, and simplicial complex partitions , 1996, Electron. J. Comb..

[15]  Bernardetta Addis,et al.  Efficiently packing unequal disks in a circle , 2008, Oper. Res. Lett..

[16]  Marco Locatelli,et al.  Packing equal circles in a square: a deterministic global optimization approach , 2002, Discret. Appl. Math..

[17]  Tibor Csendes,et al.  New Approaches to Circle Packing in a Square - With Program Codes , 2007, Optimization and its applications.

[18]  Bernardetta Addis,et al.  Disk Packing in a Square: A New Global Optimization Approach , 2008, INFORMS J. Comput..