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Convex Aggregation Problems in \(\mathbb {Z}^2\)

  • Yan GérardEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11414)

Abstract

We introduce a family of combinatorial problems of digital geometry that we call convex aggregation problems. Two variants are considered. In Unary convex aggregation problems, a first lattice set \(A\subseteq \mathbb {Z}^d\) called support and a family of lattice sets \(B^i \subseteq \mathbb {Z}^d\) called pads are given. The question to determine whether there exists a non-empty subset of pads (the set of their indices is denoted I) whose union \(A\cup _{i\in I} B ^i\) with the support is convex. In the binary convex aggregation problem, the input contains the support set \(A\subseteq \mathbb {Z}^2\) and pairs of pads \(B^i\) and \(\overline{B} ^i\). The question is to aggregate to the support either a pad \(B^i\) or its correspondent \(\overline{B} ^i\) so that the union \(A\cup _{i\in I} B ^i \cup _{i\not \in I} \overline{B} ^i\) is convex.

We provide a first classification of the classes of complexities of these two problems in dimension 2 under different assumptions: if the support is 8-connected and the pads included in its enclosing rectangle, if the pads are all disjoint, if they intersect and at least according to the chosen kind of convexity. In the polynomial cases, the algorithms are based on a reduction to Horn-SAT while in the NP-complete cases, we reduce 3-SAT to an instance of convex aggregation.

Keywords

Digital Geometry Discrete Tomography Convexity Complexity Horn-SAT 

Notes

Acknowledgement

This work has been sponsored by the French government research program “Investissements d’Avenir” through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université Clermont Auvergne - LIMOSClermont-FerrandFrance

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