Polyhredral techniques in combinatorial optimization I: theory

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems, leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done eciently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience

Polyhedral Techniques in Combinatorial Optimization

Combinatorial optimization problems arise in several areas ranging from management to mathematics and graph theory. Most combinatorial optimization problems are compu- tationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable development in polyhedral techniques leading to an increase in the size of several problem types that can be solved by a factor hundred. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. The purpose of this article is to give anoverview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also discuss several computational aspects and implementation issues related to the use of polyhedral methods.

Improved approximation algorithm for k-level UFL with penalties, a simplistic view on randomizing the scaling parameter

The state of the art in approximation algorithms for facility location problems are complicated combinations of various techniques. In particular, the currently best 1.488-approximation algorithm for the uncapacitated facility location (UFL) problem by Shi Li is presented as a result of a non-trivial randomization of a certain scaling parameter in the LP-rounding algorithm by Chudak and Shmoys combined with a primal-dual algorithm of Jain et al. In this paper we first give a simple interpretation of this randomization process in terms of solving an aux- iliary (factor revealing) LP. Then, armed with this simple view point, Abstract. we exercise the randomization on a more complicated algorithm for the k-level version of the problem with penalties in which the planner has the option to pay a penalty instead of connecting chosen clients, which results in an improved approximation algorithm

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

Given an integral $d \times n$ matrix $A$, 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 $P_A(b)=\{x: Ax=b, x\geq0\}$. 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 $P_A(b) \cap {\mathbb Z}^n$ has at least $k$ 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 $b$ for which $P_A(b) \cap {\mathbb Z}^n$ has exactly $k$ solutions or fewer than $k$ solutions. (2) A computational complexity theory. We show that, when $n$, $k$ 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 $k$ solutions. (3) Applications and computation for the $k$-Frobenius numbers. Using Generating functions we prove that for fixed $n,k$ the $k$-Frobenius number can be computed in polynomial time. This generalizes a well-known result for $k=1$ by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of $k$-Frobenius numbers and their relatives

Protected sampling is preferable in bronchoscopic studies of the airway microbiome

The aim was to evaluate susceptibility of oropharyngeal contamination with various bronchoscopic sampling techniques. 67 patients with obstructive lung disease and 58 control subjects underwent bronchoscopy with small-volume lavage (SVL) through the working channel, protected bronchoalveolar lavage (PBAL) and bilateral protected specimen brush (PSB) sampling. Subjects also provided an oral wash (OW) sample, and negative control samples were gathered for each bronchoscopy procedure. DNA encoding bacterial 16S ribosomal RNA was sequenced and bioinformatically processed to cluster into operational taxonomic units (OTU), assign taxonomy and obtain measures of diversity. The proportion of Proteobacteria increased, whereas Firmicutes diminished in the order OW, SVL, PBAL, PSB (p<0.01). The alpha-diversity decreased in the same order (p<0.01). Also, beta-diversity varied by sampling method (p<0.01), and visualisation of principal coordinates analyses indicated that differences in diversity were smaller between OW and SVL and OW and PBAL samples than for OW and the PSB samples. The order of sampling (left versus right first) did not influence alpha- or beta-diversity for PSB samples. Studies of the airway microbiota need to address the potential for oropharyngeal contamination, and protected sampling might represent an acceptable measure to minimise this problem.publishedVersio

Catecholamine Storage Vesicles: Role of Core Protein Genetic Polymorphisms in Hypertension

Hypertension is a complex trait with deranged autonomic control of the circulation. The sympathoadrenal system exerts minute-to-minute control over cardiac output and vascular tone. Catecholamine storage vesicles (or chromaffin granules) of the adrenal medulla contain remarkably high concentrations of chromogranins/secretogranins (or “granins”), catecholamines, neuropeptide Y, adenosine triphosphate (ATP), and Ca2+. Within secretory granules, granins are co-stored with catecholamine neurotransmitters and co-released upon stimulation of the regulated secretory pathway. The principal granin family members, chromogranin A (CHGA), chromogranin B (CHGB), and secretogranin II (SCG2), may have evolved from shared ancestral exons by gene duplication. This article reviews human genetic variation at loci encoding the major granins and probes the effects of such polymorphisms on blood pressure, using twin pairs to probe heritability and individuals with the most extreme blood pressure values in the population to study hypertension

Factorization of a 512 bit RSA modulus

This paper reports on the factorization of the 512 bit number RSA-155 by the number field Sieve factoring method (NFS) and discusses the implications for RS

On a Clique-Based Integer Programming Formulation of Vertex Colouring with Applications in Course Timetabling

Vertex colouring is a well-known problem in combinatorial optimisation, whose alternative integer programming formulations have recently attracted considerable attention. This paper briefly surveys seven known formulations of vertex colouring and introduces a formulation of vertex colouring using a suitable clique partition of the graph. This formulation is applicable in timetabling applications, where such a clique partition of the conflict graph is given implicitly. In contrast with some alternatives, the presented formulation can also be easily extended to accommodate complex performance indicators (``soft constraints'') imposed in a number of real-life course timetabling applications. Its performance depends on the quality of the clique partition, but encouraging empirical results for the Udine Course Timetabling problem are reported