Minkowski decomposition of convex lattice polygons

Abstract

A relatively recent area of study in geometric modeling concerns toric Bézier patches. In this line of work, several questions reduce to testing whether a given convex lattice polygon can be decomposed into a Minkowski sum of two such polygons and, if so, to finding one or all such decompositions. Other motivations for this problem include sparse resultant computation, especially for the implicitization of parametric surfaces, and factorization of bivariate polynomials. Particularly relevant for geometric modeling are decompositions where at least one summand has a small number of edges. We study the complexity of Minkowski decomposition and propose efficient algorithms for the case of constant-size summands. We have implemented these algorithms and illustrate them by various experiments with random lattice polygons and on all convex lattice polygons with zero or one interior lattice points. We also express the general problem by means of standard and well-studied problems in combinatorial optimization. This leads to an improvement in asymptotic complexity and, eventually, to efficient randomized algorithms and implementations.