A New Verified Optimization Technique For The "Packing Circles In A Unit Square" Problems

The paper presents a new verified optimization method for the problem of finding the densest packing of non-overlapping equal circles within a square. In order to provide reliable numerical results, the developed algorithm is based on interval analysis. As one of the most efficient parts of the algorithm, an interval-based version of a previous elimination procedure is introduced. This method represents the remaining areas still of interest as polygons fully calculated in a reliable way. The most promising strategy of finding optimal circle packing configurations is currently the partitioning of the original problem into subproblems. Still as a result of the highly increasing number of subproblems, earlier computer-aided methods were not able to solve problem instances where the number of circles was greater than 27. The present paper provides a carefully developed technique resolving this difficulty by eliminating large groups of subproblems together. As a demonstration of the capabilities of the new algorithm the problems of packing 28, 29, and 30 circles were solved within very tight tolerance values. Our verified procedure decreased the uncertainty in the location of the optimal packings by more than 700 orders of magnitude in all cases

A new verified optimization technique for the “packing circles in a unit square” problems

Abstract. The paper presents a new verified optimization method for the problem of finding the densest packings of non-overlapping equal circles in a square. In order to provide reliable numerical results, the developed algorithm is based on interval analysis. As one of the most efficient parts of the algorithm, an interval-based version of a previous elimination procedure is introduced. This method represents the remaining areas still of interest as polygons fully calculated in a reliable way. Currently the most promising strategy of finding optimal circle packing configurations is to partition the original problem into subproblems. Still as a result of the highly increasing number of subproblems, earlier computer-aided methods were not able to solve problem instances where the number of circles was greater than 27. The present paper provides a carefully developed technique resolving this difficulty by eliminating large groups of subproblems together. As a demonstration of the capabilities of the new algorithm the problems of packing 28, 29, and 30 circles were solved within very tight tolerance values. Our verified procedure decreased the uncertainty in the location of the optimal packings by more than 700 orders of magnitude in all cases

A reliable area reduction technique for solving circle packing problems

We are dealing with the optimal, i.e. densest packings of congruent circles into a unit square. In the recent years we built a numerically reliable method using interval arithmetic computations, which can be regarded as a `computer assisted proof'. A very efficient algorithm for eliminating large sets of suboptimal points is well known from earlier, non-interval computer methods. The paper presents an interval-based version of this tool, implemented as an accelerating device of an interval branch-and-bound optimization algorithm. In order to satisfy the requirements of a computer proof, detailed algorithmic descriptions and a proof of correctness are provided. The elimination method played a key role in solving the earlier open problems of packing 28, 29, and 30 circles