Constrained infinite group relaxations of MIPs

Recently minimal and extreme inequalities for continuous group relaxations of general mixed integer sets have been characterized. In this paper, we consider a stronger relaxation of general mixed integer sets by allowing constraints, such as bounds, on the free integer variables in the continuous group relaxation. We generalize a number of results for the continuous infinite group relaxation to this stronger relaxation and characterize the extreme inequalities when there are two integer variables.

Some lower bounds on sparse outer approximations of polytopes

Motivated by the need to better understand the properties of sparse cutting-planes used in mixed integer programming solvers, the paper [2] studied the idealized problem of how well a polytope is approximated by the use of sparse valid inequalities. As an extension to this work, we study the following less idealized questions in this paper: (1) Are there integer programs, such that sparse inequalities do not approximate the integer hull well even when added to a linear programming relaxation? (2) Are there polytopes, where the quality of approximation by sparse inequalities cannot be significantly improved by adding a budgeted number of arbitrary (possibly dense) valid inequalities? (3) Are there polytopes that are difficult to approximate under every rotation? (4) Are there polytopes that are difficult to approximate in all directions using sparse inequalities? We answer each of the above questions in the positive

Split rank of triangle and quadrilateral inequalities

A simple relaxation of two rows of a simplex tableau is a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. Recently Andersen et al. [2] and Cornu´ejols and Margot [13] showed that the facet-defining inequalities of this set are either split cuts or intersection cuts obtained from lattice-free triangles and quadrilaterals. Through a result by Cook et al. [12], it is known that one particular class of facet- defining triangle inequality does not have a finite split rank. In this paper, we show that all other facet-defining triangle and quadrilateral inequalities have finite split rank. The proof is constructive and given a facet-defining triangle or quadrilateral inequality we present an explicit sequence of split inequalities that can be used to generate it.mixed integer programs, split rank, group relaxations

Mixed-integer Quadratic Programming is in NP

Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP and integer linear programming is in NP

Integer quantum Hall effect in a square lattice revisited

We investigate the phenomenon of integer quantum Hall effect in a square lattice, subjected to a perpendicular magnetic field, through Landauer-B\"uttiker formalism within the tight-binding framework. The oscillating nature of longitudinal resistance and near complete suppression of momentum relaxation processes are examined by studying the flow of charge current using Landauer-Keldysh prescription. Our analysis for the lattice model corroborates the finding obtained in the continuum model and provides a simple physical understanding.Comment: 5 pages, 5 figure

Effect of Dephasing on Electron Transport in a Molecular Wire: Green's Function Approach

The effect of dephasing on electron transport through a benzene molecule is carefully examined using a phenomenological model introduced by B\"{u}ttiker. Within a tight-binding framework all the calculations are performed based on the Green's function formalism. We investigate the influence of dephasing on transmission probability and current-voltage characteristics for three different configurations ({\em ortho}, {\em meta} and {\em para}) of the molecular system depending on the locations of two contacting leads. The presence of dephasing provides a significant change in the spectral properties of the molecule and exhibits several interesting patterns that have so far remain unexplored.Comment: 8 pages, 10 figure