Online Square Packing
Sandor P. Fekete, Tom Kamphans, Nils Schweer

TL;DR
This paper studies the online problem of packing squares into a strip with gravity constraints, introducing new algorithms with improved competitive ratios over previous methods.
Contribution
It presents a geometric analysis and new algorithms achieving better competitive ratios for online square packing with gravity constraints.
Findings
Achieved a 3.5-competitive ratio with the bottom-left heuristic.
Developed a 2.615-competitive algorithm for the problem.
Extended previous work by incorporating gravity into the analysis.
Abstract
We analyze the problem of packing squares in an online fashion: Given a semi-infinite strip of width 1 and an unknown sequence of squares of side length in [0,1] that arrive from above, one at a time. The objective is to pack these items as they arrive, minimizing the resulting height. Just like in the classical game of Tetris, each square must be moved along a collision-free path to its final destination. In addition, we account for gravity in both motion (squares must never move up) and position (any final destination must be supported from below). A similar problem has been considered before; the best previous result is by Azar and Epstein, who gave a 4-competitive algorithm in a setting without gravity (i.e., with the possibility of letting squares "hang in the air") based on ideas of shelf-packing: Squares are assigned to different horizontal levels, allowing an analysis that is…
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Taxonomy
TopicsOptimization and Search Problems · Optimization and Packing Problems · Computability, Logic, AI Algorithms
