Slack-Pack algorithm for Meir-Moser packing problem
A. D. Kislovskiy, E. Yu. Lerner, I. A. Senkevich

TL;DR
This paper introduces the Slack-Pack algorithm, a novel tiling method for the Meir-Moser problem, which manages gaps and controls empty space ratios, extending Tao's recent results on tiling with scaled rectangles.
Contribution
The paper presents the Slack-Pack algorithm, enabling controlled tiling with gaps, and provides arguments supporting the extension of Tao's tiling results to the case t=1.
Findings
The Slack-Pack algorithm effectively manages gaps in tiling.
It controls the ratio of empty space to total area.
The method supports extending Tao's results to t=1.
Abstract
The well-known problem stated by A. Meir and L. Moser consists in tiling the unit square with rectangles (details), whose side lengths equal , where indices~ range from 1 to infinity. Recently, Terence Tao has proved that it is possible to tile with rectangles (squares with the side length of ), , the square, whose area equals the sum of areas of these details, provided that only those details, whose indices exceed certain~, are taken into consideration. We adduce arguments in favor of the assumption that the result obtained by T. Tao is also valid for . We use a new tiling method (the Slack-Pack algorithm), which initially admits gaps between stacks of details. The algorithm uses a pre-fixed parameter , , connected with the gap value. The new algorithm allows one to control the ratio…
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Taxonomy
TopicsOptimization and Packing Problems · Advanced Manufacturing and Logistics Optimization · VLSI and FPGA Design Techniques
