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.Key words. interval arithmetic, branch-and-bound method, circle packing, optimality proof AMS subject classifications. 52C15, 52C26, 65G30, 65K05, 90C301. Introduction. The so-called 'optimal packing of equal circles into a square' problem class is an interesting part of geometrical optimization. Circle packing has several real-life applications, e.g. cutting out a given number of identical circle shaped objects from some kind of material with a minimal amount of waste. In addition, proving the optimality of a packing configuration is a serious theoretical challenge from both the mathematical and the computational points of view.The paper is organized as follows: in §2 we discuss the general definition of the problem and give a brief historical overview focusing mostly on those of the computeraided methods which form the basis of the present algorithm. Section 3 contains the basic definitions and properties of interval analysis. In §4 a general interval branchand-bound frame algorithm is introduced. After that, crucial parts of the B&B algorithm are discussed in detail, as those designed specifically for circle packing problems. As a specification step, §5 introduces a new elimination procedure based on a prior non-interval one known from the literature. In §6 the questions of finding global solutions are discussed and new methods for eliminating tile combinations are investigated. In §7 we propose an efficient way of handling occurrences of free (not fixed) circles in optimal packings. Finally, in §8 the results of the previous sections are applied by solving the packing problems of 28, 29, and 30 circles. The presented numerical results demonstrate how the algorithm works in the successive elimination steps.
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