Minimizing the number of lattice points in a translated polygon
نویسندگان
چکیده
The parametric lattice-point counting problem is as follows: Given an integer matrix A ∈ Zm×n , compute an explicit formula parameterized by b ∈ R that determines the number of integer points in the polyhedron {x ∈R : Ax É b}. In the last decade, this counting problem has received considerable attention in the literature. Several variants of Barvinok’s algorithm have been shown to solve this problem in polynomial time if the number n of columns of A is fixed. Central to our investigation is the following question: Can one also efficiently determine a parameter b such that the number of integer points in {x ∈R : Ax É b} is minimized? Here, the parameter b can be chosen from a given polyhedron Q ⊆ R . Our main result is a proof that finding such a minimizing parameter is N P-hard, even in dimension 2 and even if the parametrization reflects a translation of a 2-dimensional convex polygon. This result is established via a relationship of this problem to arithmetic progressions and simultaneous Diophantine approximation. On the positive side we show that in dimension 2 there exists a polynomial time algorithm for each fixed k that either determines a minimizing translation or asserts that any translation contains at most 1+1/k times the minimal number of lattice points.
منابع مشابه
تعبیه ی هندسی درخت درنقاط داخل یک چندضلعی با حداقل تعداد خم
In this paper we consider to embed a tree T with N vertices on a set of N points inside a simple polygon on n vertices and the goal is to minimize the number of bends. The main idea of our algorithm is modeling the problem into graph matching problem and uses the graph matching algorithms. We apply the concept of error-correction transformation and find the appropriate cost function then we per...
متن کاملA Fast Algorithm for Covering Rectangular Orthogonal Polygons with a Minimum Number of r-Stars
Introduction This paper presents an algorithm for covering orthogonal polygons with minimal number of guards. This idea examines the minimum number of guards for orthogonal simple polygons (without holes) for all scenarios and can also find a rectangular area for each guards. We consider the problem of covering orthogonal polygons with a minimum number of r-stars. In each orthogonal polygon P,...
متن کاملOn convex lattice polygons
Let II be a convex lattice polygon with b boundary points and c (5 1) interior points. We show that for any given a , the number b satisfies b 5 2e + 7 , and identify the polygons for which equality holds. A lattice polygon II is a simple polygon whose vertices are points of the integral lattice. We let A = 4(11) denote the area of II , b{U) the number of lattice points on the boundary of II , ...
متن کاملA Short Proof of the Twelve Points Theorem
We present a short elementary proof of the following Twelve Points Theorem: Let M be a convex polygon with vertices at the lattice points, containing a single lattice point in its interior. Denote by m (resp. m) the number of lattice points in the boundary of M (resp. in the boundary of the dual polygon). Then
متن کامل