Minkowski decomposition of convex lattice polygons Ioannis Z. Emiris and Elias P. Tsigaridas Department of Informatics and Telecommunications National University of Athens, HELLAS {emiris,et}@di.uoa.gr Summary. A relatively recent area of study in geometric modelling concerns toric B´ ezier 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 implicitiza- tion of parametric surfaces, and factorization of bivariate polynomials. Particularly relevant for geometric modelling are decompositions where at least one summand has a small number of edges. We study the complexity of Minkowski decomposi- tion 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 inte- rior 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. 1 Introduction In this paper we study the decomposition of convex polygons with integral vertices (also called lattice polygons) under the Minkowski sum, which is defined as follows: Definition 1. For any two subsets A and B in Z 2 , their Minkowski sum is A ⊕ B = {a + b|a ∈ A, b ∈ B}. We call A and B the summands of A ⊕ B. The definition of the Minkowski sum can be generalized to arbitrary dimen- sion. The decomposition problem has a great interest on its own. The recent work on toric B´ ezier patches (e.g [7, 12, 13]), in geometric modelling, moti- vates several questions around this problem, mainly testing whether a given lattice polygon can be written as a Minkowski sum of two such polygons and,