720 research outputs found

    A polyhedral Frobenius theorem with applications to integer optimization

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    We prove a representation theorem of projections of sets of integer points by an integer matrix WW. Our result can be seen as a polyhedral analogue of several classical and recent results related to the Frobenius problem. Our result is motivated by a large class of nonlinear integer optimization problems in variable dimension. Concretely, we aim to optimize f(Wx)f(Wx) over a set F=P∩Zn\mathcal{F} = P\cap \mathbb{Z}^n, where ff is a nonlinear function, P⊂RnP\subset \mathbb{R}^n is a polyhedron, and W∈Zd×nW\in \mathbb{Z}^{d\times n}. As a consequence of our representation theorem, we obtain a general efficient transformation from the latter class of problems to integer linear programming. Our bounds depend polynomially on various important parameters of the input data leading, among others, to first polynomial time algorithms for several classes of nonlinear optimization problems. Read More: http://epubs.siam.org/doi/10.1137/14M097369

    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

    A generalization of the integer linear infeasibility problem

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    Does a given system of linear equations with nonnegative constraints have an integer solution? This is a fundamental question in many areas. In statistics this problem arises in data security problems for contingency table data and also is closely related to non-squarefree elements of Markov bases for sampling contingency tables with given marginals. To study a family of systems with no integer solution, we focus on a commutative semigroup generated by a finite subset of Zd\Z^d and its saturation. An element in the difference of the semigroup and its saturation is called a ``hole''. We show the necessary and sufficient conditions for the finiteness of the set of holes. Also we define fundamental holes and saturation points of a commutative semigroup. Then, we show the simultaneous finiteness of the set of holes, the set of non-saturation points, and the set of generators for saturation points. We apply our results to some three- and four-way contingency tables. Then we will discuss the time complexities of our algorithms.Comment: This paper has been published in Discrete Optimization, Volume 5, Issue 1 (2008) p36-5

    Semistable reduction for overconvergent F-isocrystals, III: Local semistable reduction at monomial valuations

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    We resolve the local semistable reduction problem for overconvergent F-isocrystals at monomial valuations (Abhyankar valuations of height 1 and residue transcendence degree 0). We first introduce a higher-dimensional analogue of the generic radius of convergence for a p-adic differential module, which obeys a convexity property. We then combine this convexity property with a form of the p-adic local monodromy theorem for so-called fake annuli.Comment: 36 pages; v3: refereed version; adds appendix with two example

    Forall-exist statements in pseudopolynomial time

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    Given a convex set Q⊆RmQ \subseteq R^m and an integer matrix W∈Zm×nW \in Z^{m \times n}, we consider statements of the form ∀b∈Q∩Zm \forall b \in Q \cap Z^m ∃x∈Zn\exists x \in Z^n s.t. Wx≤bWx \leq b. Such statements can be verified in polynomial time with the algorithm of Kannan and its improvements if nn is fixed and QQ is a polyhedron. The running time of the best-known algorithms is doubly exponential in~nn. In this paper, we provide a pseudopolynomial-time algorithm if mm is fixed. Its running time is (mΔ)O(m3)(m \Delta)^{O(m^3)}, where Δ=∥W∥∞\Delta = \|W\|_\infty. Furthermore it applies to general convex sets QQ. Second, we provide new upper bounds on the \emph{diagonal} as well as the \emph{polyhedral Frobenius} number, two recently studied forall-exist problems

    The distributions of functions related to parametric integer optimization

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    We consider the asymptotic distribution of the IP sparsity function, which measures the minimal support of optimal IP solutions, and the IP to LP distance function, which measures the distance between optimal IP and LP solutions. We create a framework for studying the asymptotic distribution of general functions related to integer optimization. There has been a significant amount of research focused around the extreme values that these functions can attain, however less is known about their typical values. Each of these functions is defined for a fixed constraint matrix and objective vector while the right hand sides are treated as input. We show that the typical values of these functions are smaller than the known worst case bounds by providing a spectrum of probability-like results that govern their overall asymptotic distributions.Comment: Accepted for journal publicatio

    Presburger arithmetic, rational generating functions, and quasi-polynomials

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    Presburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, if p=(p_1,...,p_n) are a subset of the free variables in a Presburger formula, we can define a counting function g(p) to be the number of solutions to the formula, for a given p. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.Comment: revised, including significant additions explaining computational complexity results. To appear in Journal of Symbolic Logic. Extended abstract in ICALP 2013. 17 page
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