Quantitative Combinatorial Geometry for Continuous Parameters

We prove variations of Carathéodory’s, Helly’s and Tverberg’s theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lovász’s colorful Helly’s theorem, Bárány’s colorful Carathéodory’s theorem, and the colorful Tverberg’s theorem

The flip-graph of the 4-dimensional cube is connected

Flip-graph connectedness is established here for the vertex set of the 4-dimensional cube. It is found as a consequence that this vertex set has 92 487 256 triangulations, partitioned into 247 451 symmetry classes.Comment: 20 pages, 3 figures, revised proofs and notation

Alimentación práctica del cerdo

The feed represent over 65% of production costs, so should be established as a priority. It is not enough that a diet meets the nutritional needs of pigs, the ration formulation must right with official regulations governing each country for the use and manufacture of feed. Also, the feed should be easy to preserve and supplying, taking into the wide variety to installations (feeders and drinkers) used in various stages of pigs. However, the fundamental objective of the formulation of a diet is that it contains the necessary nutrients in the correct proportions and balance, considering the physiological stage, weight, age, sex, genetic potential, health status, season and production objectives with a the legal constraints. Once accomplished the formulation, the next step will be insure preparing the feed under conditions that ensure the safety, traceability and lower cost of the same. To this challenge, is adding the need to the right with environmental regulations related to feed and animal welfare.La alimentación representan alrededor del 65% de los costes de producción, por ello debe establecerse como una prioridad. No es suficiente que una dieta cumpla con las necesidades nutricionales de los cerdos, la formulación debe obedecer las normativas oficiales que rigen en cada país para el uso y fabricación de alimentos. Asimismo, el alimento debe ser fácil de conservar y suministrar, asumiendo la gran variedad de instalaciones (comederos y bebederos) utilizadas en las distintas etapas de los cerdos. Sin embargo, el objetivo fundamental de la formulación de una dieta es que contenga los nutrientes necesarios en las cantidades correctas y equilibradas, considerando la etapa fisiológica, peso, edad, sexo, potencial genético, estado de salud, época del año, objetivos productivos y de producto final, así como las limitantes legales. Una vez cumplida la formulación, el siguiente paso es asegurar que ésta sea elaborada bajo condiciones que garanticen la inocuidad, trazabilidad y bajo costo de la misma. A este desafío, se añade la necesidad de cumplir con las normativas ambientales relacionadas con la alimentación y bienestar animal

Model counting for complex data structures

We extend recent approaches for calculating the probability of program behaviors, to allow model counting for complex data structures with numeric fields. We use symbolic execution with lazy initialization to compute the input structures leading to the occurrence of a target event, while keeping a symbolic representation of the constraints on the numeric data. Off-the-shelf model counting tools are used to count the solutions for numerical constraints and field bounds encoding data structure invariants are used to reduce the search space. The technique is implemented in the Symbolic PathFinder tool and evaluated on several complex data structures. Results show that the technique is much faster than an enumeration-based method that uses the Korat tool and also highlight the benefits of using the field bounds to speed up the analysis

Polyhedra Circuits and Their Applications

To better compute the volume and count the lattice points in geometric objects, we propose polyhedral circuits. Each polyhedral circuit characterizes a geometric region in Rd . They can be applied to represent a rich class of geometric objects, which include all polyhedra and the union of a finite number of polyhedron. They can be also used to approximate a large class of d-dimensional manifolds in Rd . Barvinok [3] developed polynomial time algorithms to compute the volume of a rational polyhedron, and to count the number of lattice points in a rational polyhedron in Rd with a fixed dimensional number d. Let d be a fixed dimensional number, TV(d,n) be polynomial time in n to compute the volume of a rational polyhedron, TL(d,n) be polynomial time in n to count the number of lattice points in a rational polyhedron, where n is the total number of linear inequalities from input polyhedra, and TI(d,n) be polynomial time in n to solve integer linear programming problem with n be the total number of input linear inequalities. We develop algorithms to count the number of lattice points in geometric region determined by a polyhedral circuit in O(nd⋅rd(n)⋅TV(d,n)) time and to compute the volume of geometric region determined by a polyhedral circuit in O(n⋅rd(n)⋅TI(d,n)+rd(n)TL(d,n)) time, where rd(n) is the maximum number of atomic regions that n hyperplanes partition Rd . The applications to continuous polyhedra maximum coverage problem, polyhedra maximum lattice coverage problem, polyhedra (1−β) -lattice set cover problem, and (1−β) -continuous polyhedra set cover problem are discussed. We also show the NP-hardness of the geometric version of maximum coverage problem and set cover problem when each set is represented as union of polyhedra

Verifying integer programming results

Software for mixed-integer linear programming can return incorrect results for a number of reasons, one being the use of inexact floating-point arithmetic. Even solvers that employ exact arithmetic may suffer from programming or algorithmic errors, motivating the desire for a way to produce independently verifiable certificates of claimed results. Due to the complex nature of state-of-the-art MIP solution algorithms, the ideal form of such a certificate is not entirely clear. This paper proposes such a certificate format designed with simplicity in mind, which is composed of a list of statements that can be sequentially verified using a limited number of inference rules. We present a supplementary verification tool for compressing and checking these certificates independently of how they were created. We report computational results on a selection of MIP instances from the literature. To this end, we have extended the exact rational version of the MIP solver SCIP to produce such certificates