Review of Combinatorial Optimization - Theory and Algorithms

Abstract

Rarely do mathematical disciplines have so direct practical relevance as combinatorial optimizations, which is why it is one of the most active areas of discrete mathematics. It became a subject in its own right only about 50 years ago, which makes it one of the youngest also.This book covers most of the important results and algorithms achieved in the field to date. Most of the problems are formulated in terms of graphs and linear programs. The book starts with reviewing basic graph theory and linear and integer programming. Next, the classical topics in the field are studied: minimum spanning trees, shortest paths, network flows, matching and matroids.Most of the problems in Chapters 6-14 have polynomial time (efficient) algorithms, while most of the problems studied in Chapters 15-21 are NP-hard, i.e. polynomial time algorithm is unlikely to exist. Although in many cases approximation algorithms are offered which at least have guaranteed performance.Some of the topics include areas which have developed very recently, and which have not appeared in a book before. Examples are algorithms for multicommodity flows, network design problems and the traveling salesman problem. The book also contains some new results and new proofs for previously known results.The authors have an unique approach in the presentation of the topics. They provided detailed proofs for almost all results, including deep classical theorems (e.g. weighted matching algorithm and Karmarkar-Karp bin-packing algorithm) whose proofs are usually sketched in previous works.