Antoniadis, A.

Biermeier, F.

Cristi, A.

Damerius, C.

Hoeksma, R.

Kaaser, D.

Kling, P.

Nölke, L.

English

MPG.PuRe

On the Complexity of Anchored Rectangle Packing

In the Anchored Rectangle Packing (ARP) problem, we are given a set of points P in the unit square [0, 1]2 and seek a maximum-area set of axis-aligned interior-disjoint rectangles S, each of which is anchored at a point p ∈ P. In the most prominent variant – Lower-Left-Anchored Rectangle Packing (LLARP) – rectangles are anchored in their lower-left corner. Freedman [19, Unsolved Problem 11, page 345] conjectured in 1969 that, if (0, 0) ∈ P, then there is a LLARP that covers an area of at least 0.5. Somewhat surprisingly, this conjecture remains open to this day, with the best known result covering an area of 0.091 [11]. Maybe even more surprisingly, it is not known whether LLARP – or any ARP-problem with only one anchor – is NP-hard. In this work, we first study the Center-Anchored Rectangle Packing (CARP) problem, where rectangles are anchored in their center. We prove NP-hardness and provide a PTAS. In fact, our PTAS applies to any ARP problem where the anchor lies in the interior of the rectangles. Afterwards, we turn to the LLARP problem and investigate two different resource-augmentation settings: In the first we allow an ε-perturbation of the input P, whereas in the second we permit an ε-overlap between rectangles. For the former setting, we give an algorithm that covers at least as much area as an optimal solution of the original problem. For the latter, we give an (1 − ε)-approximation.</p

Antoniadis, Antonios

Cristi, Andrés

Hoeksma, Ruben

Kling, Peter

Biermeier, Felix

Damerius, Christoph

Kaaser, Dominik

Nölke, Lukas

University of Twente Research Information

On the complexity of anchored rectangle packing

Dagstuhl Research Online Publication Server

https://research.utwente.nl/en/publications/2371c637-3008-4234-b90f-226d2607c4fc

On the Complexity of Anchored Rectangle Packing

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MPG.PuRe

University of Twente Research Information

Dagstuhl Research Online Publication Server