Inhaltsverzeichnis

Part I (by Bernd Gärtner): 1 Introduction: MAXCUT via Semidefinite Programming.- 2 Semidefinite Programming.- 3 The Shannon Capacity and the Lovász Theta.- 4 Duality and Cone Programming.- 5 Approximately Solving Semidefinite Programs.- 6 An Interior-Point Algorithm for Semidefinite Programming.- 7 Copositive Programming.- Part II (by Jiri Matousek): 8 Lower Bounds for the Goemans Williamson MAXCUT Algorithm .- 9 Coloring 3-Chromatic Graphs.- 10 Maximizing a Quadratic Form on a Graph.- 11 Colorings With Low Discrepancy.- 12 Constraint Satisfaction Problems, and Relaxing Them Semidefinitely.- 13 Rounding Via Miniatures.- Summary.- References.- Index

Annotation

This introduction to aspects of semidefinite programming and its use in approximation algorithms develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others.

Beschreibung

Semidefinite programs constitute one of the largest classes of optimization problems that can be solved with reasonable efficiency both in theory and practice and they play a key role in a variety of research areas, such as combinatorial optimization, approximation algorithms, computational complexity, graph theory, geometry, real algebraic geometry and quantum computing. This book is an introduction to selected aspects of semidefinite programming and its use in approximation algorithms. It covers the basics but also a significant amount of recent and more advanced material.
There are many important computational problems, such as MAXCUT, for which it is not reasonable to expect to obtain an exact solution efficiently. In such cases, one has to settle for approximate solutions. For MAXCUT and its relatives, there has been a development that reveals that algorithms based on semidefinite programming deliver the best possible approximation ratio among all polynomial-time algorithms. It is tentative since it relies on the Unique Games Conjecture, which is as yet unproven, but for those who believe in that conjecture, semidefinite programming is the ultimate tool for these problems. This book follows the semidefinite side of these developments, presenting some of the main ideas behind approximation algorithms based on semidefinite programming. It develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others. It also includes applications, focusing on approximation algorithms.