An Algorithmic Theory of Numbers, Graphs, and Convexity

An Algorithmic Theory of Numbers, Graphs, and Convexity

by Laszlo Lovasz
     
 

ISBN-10: 0898712033

ISBN-13: 9780898712032

Pub. Date: 01/28/1986

Publisher: SIAM

A study of how complexity questions in computing interact with classical mathematics in the numerical analysis of issues in algorithm design. Algorithmic designers concerned with linear and nonlinear combinatorial optimization will find this volume especially useful. Two algorithms are studied in detail: the ellipsoid method and the simultaneous diophantine

Overview

A study of how complexity questions in computing interact with classical mathematics in the numerical analysis of issues in algorithm design. Algorithmic designers concerned with linear and nonlinear combinatorial optimization will find this volume especially useful. Two algorithms are studied in detail: the ellipsoid method and the simultaneous diophantine approximation method. Although both were developed to study, on a theoretical level, the feasibility of computing some specialized problems in polynomial time, they appear to have practical applications. The book first describes use of the simultaneous diophantine method to develop sophisticated rounding procedures. Then a model is described to compute upper and lower bounds on various measures of convex bodies. Use of the two algorithms is brought together by the author in a study of polyhedra with rational vertices. The book closes with some applications of the results to combinatorial optimization.

Product Details

ISBN-13:
9780898712032
Publisher:
SIAM
Publication date:
01/28/1986
Series:
CBMS-NSF Regional Conference Series in Applied Mathematics
Pages:
91
Product dimensions:
5.98(w) x 8.98(h) x 0.24(d)

Table of Contents

How to Round Numbers; Preliminaries: On Algorithms Involving Numbers; Diophantine Approximation, Problems; Lattices, Bases, and the Reduction Problem; Diophantine Approximation and Rounding; What is a Real Number How to Round a Convex Body; Preliminaries: Inputting a Set; Algorithmic Problems on Convex Sets; The Ellipsoid Method; Rational Polyhedra; Some Other Algorithmic Problems on Convex Sets; Integer Programming in Fixed Dimension; Some Applications in Combinatorics; Cuts and Joins; Chromatic Number, Cliques and Perfect Graphs; Minimizing a Submodular Function.

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