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Approximation algorithms. (English) Zbl 1005.68002

Berlin: Springer. xix, 378 p. (1999).
Approximation algorithms is an area where much progress has been made in the last 10 years. The book under review is a very good help for understanding these results.
In each of the first 27 chapters an important combinatorial optimization problem is presented and one or more approximation algorithms for it are clearly and concisely described and analyzed. In this way most of the most important results from the approximation algorithm literature are covered, often more easily comprehensible than the original articles.
The book consists of three parts. In the first part combinatorial approximation algorithms for a dozen of problems such as Set cover, Knapsack and Euclidean TSP are presented. In the second part linear programming based approximation algorithms for a dozen of problems such as Set cover, Maximum satisfiability and Facility location are presented. The last part consists of one chapter giving approximation algorithms for the Shortest vector problem, one chapter giving approximation algorithms for counting problems and one chapter dealing with approximation hardness results. The book ends with a list of open problems and appendices on complexity and probability theory.
I think that the book is suitable for a graduate course on approximation algorithms, but also as a reference for researchers in the area. It is probably too hard to be used in an under graduate course.
This is the third book ever published dedicated to approximation. The first book was D. S. Hochbaum [Approximation Algorithms for NP-Hard Problems, Boston, Mass., PWS Publ. (1997)] and consists of 13 separate chapters, each written by a leading experts in some subarea of approximation. More different algorithms for the same problem and variations of the same problem are presented than in Vazirani’s book. On the other hand Vazirani’s book covers the whole area of approximation algorithms better and in a more uniform way.
The second book on approximation is [G. Ausiello et al., Complexity and Approximation – Combinatorial optimization problems and their approximability properties (Springer, Berlin) (1999; Zbl 0937.68002)] and is not structured around the problems as Hochbaum and Vazirani. Instead it presents the area topic by topic, for example approximation algorithm construction methods, probabilistic approximation algorithms and heuristics. It also treats approximation complexity classes and approximation hardness results in more depth than Vazirani’s book. I think the books complement each other quite well.

MSC:

68-02 Research exposition (monographs, survey articles) pertaining to computer science
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
68W25 Approximation algorithms
68Q25 Analysis of algorithms and problem complexity
90C27 Combinatorial optimization
90C59 Approximation methods and heuristics in mathematical programming

Citations:

Zbl 0937.68002
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