Approximation algorithms for NP-complete problems on planar graphs
Journal of the ACM (JACM)
Approximation schemes for covering and packing problems in image processing and VLSI
Journal of the ACM (JACM)
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Simple planar graph partition into three forests
Discrete Applied Mathematics
Schematization of road networks
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Some optimal inapproximability results
Journal of the ACM (JACM)
Approximation algorithms for spreading points
WAOA'04 Proceedings of the Second international conference on Approximation and Online Algorithms
Nearest-neighbor searching under uncertainty
PODS '12 Proceedings of the 31st symposium on Principles of Database Systems
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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 also 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 inapproximability results for general graphs. For the case of trees and planar graphs, we give approximation algorithms whose performance depends upon the shape of the given regions and the set of orientations. When the orientations to consider are the ones given by the axes and the regions are axis-parallel rectangles, we obtain a polynomial time approximation scheme.