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Approximation Algorithms for Aligning Points

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Abstract

We study the problem of aligning as many points as possible horizontally, vertically, or diagonally, when each point is allowed to be placed anywhere in its own given region. Different shapes of placement regions and different sets of alignment orientations are considered. More generally, we assume that a graph is given on the points, and only the alignments of points that are connected in the graph count. We show that for planar graphs the problem is NP-hard, and we provide an inapproximability result for general graphs. For the case of trees and planar graphs, we give approximation algorithms whose performance depends on the shape of the given regions and the set of orientations. When the orientations are the ones given by the axes and the regions are axis-parallel rectangles, we obtain a polynomial time approximation scheme.

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Authors and Affiliations

  1. Institute of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands

    Sergio Cabello & Marc van Kreveld

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  1. Sergio Cabello
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  2. Marc van Kreveld
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Correspondence to Sergio Cabello or Marc van Kreveld.

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Communicated by B. Chazelle

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Cabello, S., van Kreveld, M. Approximation Algorithms for Aligning Points. Algorithmica 37, 211–232 (2003). https://doi.org/10.1007/s00453-003-1033-6

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  • DOI: https://doi.org/10.1007/s00453-003-1033-6

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