Approximating Multidimensional Subset Sum and Minkowski Decomposition of Polygons

Abstract

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, a multiset S of k-dimensional vectors is given, along with a target vector t, and one must decide whether there exists a subset of S that sums up to t. We prove, through a gap-preserving reduction from Set Cover that, for general dimension k, the corresponding optimization problem kD-SS-opt is not in APX, although the classic 1D-SS-opt has a PTAS. Our approach relates kD-SS with the well studied closest vector problem. On the positive side, we present a \(O(n^3/\epsilon ^2)\) approximation algorithm for 2D-SS-opt, where n is the cardinality of the multiset and \(\epsilon >0\) bounds the additive error in terms of some property of the input. We state two variations of this algorithm, which are more suitable for implementation. Employing a reduction of the optimization version of MinkDecomp to 2D-SS-opt we approximate the former: For an input polygon Q and parameter \(\epsilon >0\), we compute summand polygons A and B, where \(Q' = A+B\) is such that some geometric function differs on Q and \(Q'\) by \(O(\epsilon \, D)\), where D is the diameter of Q, or the Hausdorff distance between Q and \(Q'\) is also in \(O(\epsilon \, D)\). We conclude with experimental results based on our implementations.