4,031 research outputs found

    Basic Polyhedral Theory

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    This is a chapter (planned to appear in Wiley's upcoming Encyclopedia of Operations Research and Management Science) describing parts of the theory of convex polyhedra that are particularly important for optimization. The topics include polyhedral and finitely generated cones, the Weyl-Minkowski Theorem, faces of polyhedra, projections of polyhedra, integral polyhedra, total dual integrality, and total unimodularity.Comment: 14 page

    Parametric Polyhedra with at least kk Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

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    Given an integral d×nd \times n matrix AA, the well-studied affine semigroup \mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be stratified by the number of lattice points inside the parametric polyhedra PA(b)={x:Ax=b,x≥0}P_A(b)=\{x: Ax=b, x\geq0\}. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{ Sg}(A) such that PA(b)∩ZnP_A(b) \cap {\mathbb Z}^n has at least kk solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors bb for which PA(b)∩ZnP_A(b) \cap {\mathbb Z}^n has exactly kk solutions or fewer than kk solutions. (2) A computational complexity theory. We show that, when nn, kk are fixed natural numbers, one can compute in polynomial time an encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least kk solutions. (3) Applications and computation for the kk-Frobenius numbers. Using Generating functions we prove that for fixed n,kn,k the kk-Frobenius number can be computed in polynomial time. This generalizes a well-known result for k=1k=1 by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of kk-Frobenius numbers and their relatives

    Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

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    We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in R2\mathbb{R}^2. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in R2\mathbb{R}^2 can be done in polynomial time, while optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials. Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in R2\mathbb{R}^2 is solvable in polynomial time. We show that this holds for polynomials that can be translated into homogeneous polynomials, even when the translation vector is unknown. We demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input, thus requiring a compact representation of solutions for a general polynomial time algorithm for the unbounded case
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