Computers and Intractability

The focus of this book is to teach the reader how to identify, deal with, and understand the essence of NP-complete problems; Computers and Intractability does all of those things effectively. In a readable yet mathematically rigorous manner, the book covers topics such as how to prove that a given problem is NP-complete and how to cope with NP-complete problems. (There is even a chapter on advanced topics, with numerous references.) Computers and Intractability also contains a list of more than 300 problems--most of which are known to be NP-complete--with comments and references.

A first well written and systematic account on the hardness of this problems is the 1979 book on the theory of NP completeness by Michael R. Garey and David S. Johnson: Computers and Intractability, A Guide to the Theory of NP-Completeness (W.H. Freeman and Company, San Francisco). It is amazing how, after all these years, this book remains a fundamental one to be introduced on what can be effectively and efficiently solved by computers and above all on what it seems not efficiently solvable, independently of the advances of technology. Other texts have been published after that one, as for example the recent clear and complete overview on what has been done and extensively researched since then that has been given by Christos H. Papadimitriou in his book Computational Complexity (Addison-Wesley, 1994). Nevertheless, the Garey-Johnson book---as it is often familiarly called---remains the fundamental book for a clear introduction to this central problem of what is tractable by computers.

Starting from a very clear introduction to the technical term "NP-Complete," and to how this term gained importance for the description of the algorithmic tractability of certain problems in the early 70s, the book clearly defines, both in an intuitive and then in a formal way, what it is meant by the complexity of a problem. More than that, this complexity is directly related to the effective methods for solving problems (algorithms) and thus to computers themselves. The basic of the theory of NP-Completeness is completely covered in the first 5 chapters, beginning from a low-level introduction to some of the central notions of computational complexity and finally providing detailed definitions describing proof techniques to prove the hardness of certain problems. The remaining two chapters provide an overview on two alternative directions for further study. (The both of them have been extensively investigated in the following years.) Finally, the appendix contains more than 300 main entries on NP-Complete and NP-Hard problems, and this last part of the book is continuously referenced in journal and conference papers on the subject.

The first chapter is definitely accessible also to those that doesn't know so much mathematics, or computers related stuff, and thus the book is recommendable to those that are simply curious about the things that can be solved with computers.

But long after that course in NP theory was over, the astonishment of a different aspect of the book remains.

One course assignment was the development of 15 polynomial reduction proofs (proving the computational complexity equivalence of pairs of NP problems). Part of these proofs can be simple geometric shapes, locations and connecting lines, which are defined as elements in the 2 problems. Because the elements are rigorously defined, the resulting geometric pictures model rigorous proofs of equivalence.

I was astounded at the power of such abstractions (which most programmers perhaps wouldn't even recognize as legitimate proofs). This experience underlined the fact that rigorous logical proof may take many outer forms, whether mathematical equations, formal symbolic logic proofs such as the Irving Copi notation, or simple geometric drawings.

Many veins of rich ore may be mined from this work, and only 1 of them is NP theory. But the reader must be ready to do battle with some difficult ideas, and mathematical notation which can obscure the creativity of the material covered. (For astounding creativity, examine Cooke's Theorem proving that the Satisfiability problem is NP-Complete!)