## Section: New Results

### Approximating multidimensional subset sum and minkowski decomposition of polygons

Participants : Ioannis Emiris, Anna Karasoulou.

In [10] we consider the approximation of two NP-hard problems: Minkowski Decomposition (MinkDecomp) of lattice polygons in the plane and the closely related problem of Multidimensional Subset Sum (kD-SS) in arbitrary dimension. In kD-SS we are given an input set $S$ of k-dimensional vectors, a target vector $t$ and we ask if there exists a subset of $S$ that sums up to $t$. We prove, through a gap-preserving reduction, that, for general dimension k, kD-SS is not in APX although the classic 1D-SS is in PTAS. On the positive side, we present an $O({n}^{3}/{\u03f5}^{2})$ approximation grid based algorithm for 2D-SS, where $n$ is the cardinality of the set and $\u03f5>0$ bounds the difference of some measure of the input polygon and the sum of the output polygons. We also describe two approximation algorithms with a better experimental ratio. Applying one of these algorithms, and a transformation from MinkDecomp to 2D-SS, we can approximate Mink-Decomp. For an input polygon $Q$ and parameter $\u03f5$, we return two summands $A$ and $B$ such that $A+B={Q}^{\text{'}}$ with ${Q}^{\text{'}}$ being bounded in relation to $Q$ in terms of volume, perimeter, or number of internal lattice points, an additive error linear in and up to quadratic in the diameter of $Q$. A similar function bounds the Hausdorff distance between $Q$ and ${Q}^{\text{'}}$. We offer experimental results based on our implementation.

Joint with C. Tzovas.