A reliable area reduction technique for solving circle packing problems

Authors:
M. C. Markót;T. Csendes
Affiliations:
Advanced Concepts Team, ESTEC, European Space Agency, Noordwijk, The Netherlands;University of Szeged, Institute of Informatics, Szeged, Hungary
Venue:
Computing
Year:
2006

Citing 4
Cited 1

New computer methods for global optimization

New computer methods for global optimization
Multisection in Interval Branch-and-Bound Methods for Global Optimization II. Numerical Tests

Journal of Global Optimization
Packing equal circles in a square: a deterministic global optimization approach

Discrete Applied Mathematics
A New Verified Optimization Technique for the "Packing Circles in a Unit Square" Problems

SIAM Journal on Optimization

Interval methods for verifying structural optimality of circle packing configurations in the unit square

Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)

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Abstract

We are dealing with the optimal, i.e., densest packings of congruent circles into the unit square. In the recent years we have built a numerically reliable, verified method using interval arithmetic computations, which can be regarded as a "computer-assisted proof". An efficient algorithm has been published earlier for eliminating large sets of suboptimal points of the equivalent point packing problem. The present paper discusses an interval arithmetic based version of this tool, implemented as an accelerating device of an interval branch-and-bound optimization algorithm. In order to satisfy the rigorous requirements of a computational proof, a detailed algorithmic description and a proof of correctness are provided. This elimination method played a key role in solving the previously open problem instances of packing 28, 29, and 30 circles.