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.
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Markót, M.C., Csendes, T. A Reliable Area Reduction Technique for Solving Circle Packing Problems. Computing 77, 147–162 (2006). https://doi.org/10.1007/s00607-005-0155-x
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DOI: https://doi.org/10.1007/s00607-005-0155-x