Online Circle and Sphere Packing

In this paper we consider the Online Bin Packing Problem in three variants: Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes. The two first ones receive an online sequence of circles (items) of different radii while the third one receive an online sequence of spheres (items) of different radii, and they want to pack the items into the minimum number of unit squares, isosceles right triangles of leg length one, and unit cubes, respectively. For Online Circle Packing in Squares, we improve the previous best-known competitive ratio for the bounded space version, when at most a constant number of bins can be open at any given time, from 2.439 to 2.3536. For Online Circle Packing in Isosceles Right Triangles and Online Sphere Packing in Cubes we show bounded space algorithms of asymptotic competitive ratios 2.5490 and 3.5316, respectively, as well as lower bounds of 2.1193 and 2.7707 on the competitive ratio of any online bounded space algorithm for these two problems. We also considered the online unbounded space variant of these three problems which admits a small reorganization of the items inside the bin after their packing, and we present algorithms of competitive ratios 2.3105, 2.5094, and 3.5146 for Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes, respectively

Lower bounds for several online variants of bin packing

We consider several previously studied online variants of bin packing and prove new and improved lower bounds on the asymptotic competitive ratios for them. For that, we use a method of fully adaptive constructions. In particular, we improve the lower bound for the asymptotic competitive ratio of online square packing significantly, raising it from roughly 1.68 to above 1.75.Comment: WAOA 201

On Semantic Word Cloud Representation

We study the problem of computing semantic-preserving word clouds in which semantically related words are close to each other. While several heuristic approaches have been described in the literature, we formalize the underlying geometric algorithm problem: Word Rectangle Adjacency Contact (WRAC). In this model each word is associated with rectangle with fixed dimensions, and the goal is to represent semantically related words by ensuring that the two corresponding rectangles touch. We design and analyze efficient polynomial-time algorithms for some variants of the WRAC problem, show that several general variants are NP-hard, and describe a number of approximation algorithms. Finally, we experimentally demonstrate that our theoretically-sound algorithms outperform the early heuristics