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

## 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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