Two row mixed integer cuts via lifting

Recently, Andersen et al. [1], Borozan and Cornuéjols [6] and Cornuéjols and Margot [9] characterized extreme inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these inequalities to obtain cuts from two rows of a general simplex tableau, one approach is to extend the system to include all possible nonnegative integer variables (giving the two-row mixed integer infinite-group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we study the characteristics of these lifting functions. We begin by observing that functions giving valid coefficients for the nonnegative integer variables can be constructed by lifting a subset of the integer variables and then applying the fill-in procedure presented in Johnson [23]. We present conditions for these 'general fill-in functions" to be extreme for the two-row mixed integer infinite-group problem. We then show that there exists a unique 'trivial' lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal lattice-free triangle with integral vertices and one integer point in the relative interior of each side. In all other cases (maximal lattice-free triangle with one integer point in the relative interior of each side and non-integral vertices, and maximal lattice-free quadrilaterals), non-unique lifting functions may yield distinct extreme inequalities. For the case of a triangle with one integer point in the relative interior of each side and non-integral vertices, we present sufficient conditions to yield an extreme inequality for the two-row mixed integer infinite-group problem.

Zero-one IP problems: Polyhedral descriptions & cutting plane procedures

A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie

Light on the Infinite Group Relaxation

This is a survey on the infinite group problem, an infinite-dimensional relaxation of integer linear optimization problems introduced by Ralph Gomory and Ellis Johnson in their groundbreaking papers titled "Some continuous functions related to corner polyhedra I, II" [Math. Programming 3 (1972), 23-85, 359-389]. The survey presents the infinite group problem in the modern context of cut generating functions. It focuses on the recent developments, such as algorithms for testing extremality and breakthroughs for the k-row problem for general k >= 1 that extend previous work on the single-row and two-row problems. The survey also includes some previously unpublished results; among other things, it unveils piecewise linear extreme functions with more than four different slopes. An interactive companion program, implemented in the open-source computer algebra package Sage, provides an updated compendium of known extreme functions.Comment: 45 page

Feature Extraction and Duplicate Detection for Text Mining: A Survey

Text mining, also known as Intelligent Text Analysis is an important research area. It is very difficult to focus on the most appropriate information due to the high dimensionality of data. Feature Extraction is one of the important techniques in data reduction to discover the most important features. Proce- ssing massive amount of data stored in a unstructured form is a challenging task. Several pre-processing methods and algo- rithms are needed to extract useful features from huge amount of data. The survey covers different text summarization, classi- fication, clustering methods to discover useful features and also discovering query facets which are multiple groups of words or phrases that explain and summarize the content covered by a query thereby reducing time taken by the user. Dealing with collection of text documents, it is also very important to filter out duplicate data. Once duplicates are deleted, it is recommended to replace the removed duplicates. Hence we also review the literature on duplicate detection and data fusion (remove and replace duplicates).The survey provides existing text mining techniques to extract relevant features, detect duplicates and to replace the duplicate data to get fine grained knowledge to the user

Audio Cartography: Visual Encoding of Acoustic Parameters

Our sonic environment is the matter of subject in multiple domains which developed individual means of its description. As a result, it lacks an established visual language through which knowledge can be connected and insights shared. We provide a visual communication framework for the systematic and coherent documentation of sound in large-scale environments. This consists of visual encodings and mappings of acoustic parameters into distinct graphic variables that present plausible solutions for the visualization of sound. These candidate encodings are assembled into an application-independent, multifunctional, and extensible design guide. We apply the guidelines and show example maps that acts as a basis for the exploration of audio cartography

Exploiting parallelism in n-D convex hull algorithms

PhD ThesisThe convex hull is a problem of primary importance because of its applications in computational geometry. A number of sequential and parallel algorithms for computing the convex hull of a finite set of points in the lower dimensions are known. In compar- ison, the general n-D problem is not as well understood and parallel algorithms are not so prevalent because the 2-D and 3-D methods are not easily extended to the general case. This thesis presents parallel algorithms for evaluating the general n- D convex hull problem (where 2-D and 3-D are special cases) using Swart's sequential algorithm. One of our methods combines a gift-wrapping technique with partitioning and merge algorithms > where the original list is split into p 1 partitions followed by the computation of the subhulls using the sequential n-D gift-wrapping method. The partial hulls are then combined using a fanin tree. The second method computes the convex hull in parallel by wrapping around the edges until a complete facial lattice structure of the polytope is generated. Several parameterised versions of the proposed algorithms have been implemented on the shared memory and message passing architectures. In the former, performance on an Encore Multimax using Encore Parallel Threads and the more lightweight Microthread programming utilities are examined. In the latter, performance on a transputer based machine using CS- Tools is discussed. We have shown that our techniques will be useful in the construction of faster algorithms which employ the n-D convex hull algorithms as a sub-algorithmCommonwealth Scholarship Commission in the United Kingdo

Coordination of Multirobot Teams and Groups in Constrained Environments: Models, Abstractions, and Control Policies

Robots can augment and even replace humans in dangerous environments, such as search and rescue and reconnaissance missions, yet robots used in these situations are largely tele-operated. In most cases, the robots\u27 performance depends on the operator\u27s ability to control and coordinate the robots, resulting in increased response time and poor situational awareness, and hindering multirobot cooperation. Many factors impede extended autonomy in these situations, including the unique nature of individual tasks, the number of robots needed, the complexity of coordinating heterogeneous robot teams, and the need to operate safely. These factors can be partly addressed by having many inexpensive robots and by control policies that provide guarantees on convergence and safety. In this thesis, we address the problem of synthesizing control policies for navigating teams of robots in constrained environments while providing guarantees on convergence and safety. The approach is as follows. We first model the configuration space of the group (a space in which the robots cannot violate the constraints) as a set of polytopes. For a group with a common goal configuration, we reduce complexity by constructing a configuration space for an abstracted group state. We then construct a discrete representation of the configuration space, on which we search for a path to the goal. Based on this path, we synthesize feedback controllers, decentralized affine controllers for kinematic systems and nonlinear feedback controllers for dynamical systems, on the polytopes, sequentially composing controllers to drive the system to the goal. We demonstrate the use of this method in urban environments and on groups of dynamical systems such as quadrotors. We reduce the complexity of multirobot coordination by using an informed graph search to simultaneously build the configuration space and find a path in its discrete representation to the goal. Furthermore, by using an abstraction on groups of robots we dissociate complexity from the number of robots in the group. Although the controllers are designed for navigation in known environments, they are indeed more versatile, as we demonstrate in a concluding simulation of six robots in a partially unknown environment with evolving communication links, object manipulation, and stigmergic interactions