Customer Reviews

Combinatorial Optimization: Algorithms and Complexity (Dover Books on Computer Science)

5 star:		(14)
4 star:		(2)
3 star:		(1)
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1 star:		(0)

 

 

I had this book on my shelf for two years before taking a serious look at it, and only wish I had read it much earlier in life. Christos Papadimitriou has written quite a gem! On one hand this book serves as a good introduction to combinatorial optimization algorithms, in that it provides a flawless introduction to the simplex algorithm, linear and integer programming, and search techniques such as Branch-and-Bound and dynamic programming. On another, it serves as a good reference for many graph-theoretic algorithms. But most importantly Papadimitriou and Steiglitz seem to be on a quest to understand why some problems, such as Minimum Path or Matching, have efficient solutions, while others, such as Traveling Salesman, do not. And in doing so they end up providing the reader with a big picture behind algorithms and complexity, and the connection between optimization problems and complexity.

After reading this and Papadimitriou's "Introduction to Computational Complexity" (which I also highly recommend), I now consider him one of the best at conveying complex ideas in a way that rarely confuses the reader. I also had the priviledge of attending one of his talks on complexity, and he seems just as effusive and transparent as a lecturer as he does a writer. Ah, for once I bought a Dover book that did not disappoint.

This is just a note to mention that athough Amazon has dated this book as published in 1998, it is actually around 15 years old. By the way, it's a good book, but I didn't find it an easy read, especially the first half. One needs to already have a foundation in linear programming and optimization to digest it. A previous reviewer who said that every programmer should read it was being unduly exuberant, presumably because it happened to hit his particular spot. Most programmers don't need combinatorial optimization and for those who do there are some good alternative books.

Christos Papadimitriou, my hero is a hope for all of us who wish to master the fascinating field of Combinatorial Optimisation. Especially recommended are the chapters on matching, NP Completeness and Approximation Algorithms.

As another reader has remarked, this book is quite old though (published first in 1982). For a more to date book on Combinatorial Optimisation, one might want to look at Cook, Cunningham, Pulleyblank and Schrijver's book on Combinatorial Optimisation (published in 1998).

One could buy this book for different reasons: interests in combinatorial optimization, of course; interests in what Papadimitriou has to say, since his thoughts on this subject are definitely invaluable; perhaps the price is a good reason alone.
Whatever the reason, however, I think that would be a rare event to remain duped.

I was preparing my exam in Computability and Complexity when I first used it. I've been wonderfully surprised by the amount of definitions, algorithms, concepts I've found in this book. I think one could use this book for a simple course on Algorithms, on Computability and/or Complexity, on the whole Combinatorial Optimization, and the book would be always and costantly useful.

The chapters on algorithms and complexity, or those on NP completeness have proved to be gems. The chapters on Approximation and Local Search are great, and they feature a bunch of detailed and excellent quality stuff (e.g. there is a detailed treatment of Christofides' algorithm to approximate the TSP, that is quite an idiosyncratic topic).

All in all, a very great book, with a value exponentially greater than the very insignificant price.

This is a very nice, self-contained introduction to linear programming, algorithm design and analysis, and computational complexity. The contents are as follows:

Chap. 1 Optimization Problems 1.1 Introduction; 1.2 Optimization Problems; 1.3 Neighborhoods; 1.4 Local and Global Optima; 1.5 Convex Sets and Functions; 1.6 Convex Programming Problems

Chap. 2 The Simplex Algorithm 2.1 Forms of the Linear Programming Problem; 2.2 Basic Feasible Solutions; 2.3 The Geometry of Linear Programs; 2.3.1 Linear and Affine Spaces; 2.3.2 Convex Polytopes; 2.3.3 Polytopes and LP; 2.4 Moving from bfs to bfs; 2.5 Organization of a Tableau; 2.6 Choosing a Profitable Column; 2.7 Degeneracy and Bland's Anticycling Algorithm; 2.8 Beginning the Simplex Algorithm; 2.9 Geometric Aspects of Pivoting

Chap. 3 Duality 3.1 The Dual of a Linear Program in General Form; 3.2 Complementary Slackness; 3.3 Farkas' Lemma; 3.4 The Shortest-Path Problem and Its Dual; 3.5 Dual Information in the Tableau; 3.6 The Dual Simplex Algorithm; 3.7 Interpretation of the Dual Simplex Algorithm

Chap. 4 Computational Considerations for the Simplex Algorithm 4.1 The Revised Simplex Algorithm; 4.2 Compuational Implications of the Revised Simplex Algorithm; 4.3 The Max-Flow Problem and Its Solution by the Revised Method; 4.4 Dantzig-Wolfe Decomposition

Chap. 5 The Primal-Dual Algorithm 5.1 Introduction; 5.2 The Primal-Dual Algorithm; 5.3 Comments on the Primal-Dual Algorithm; 5.4 The Primal-Dual Method Applied to the Shortest-Path Problem; 5.5 Comments on Methodology; 5.6 The Primal-Dual Method Applied to Max-Flow

Chap. 6 Primal-Dual Algorithms for Max-Flow and Shortest Path: Ford-Fulkerson and Dijkstra 6.1 The Max-Flow, Min-Cut Theorem; 6.2 The Ford and Fulkerson Labeling Algorithm; 6.3 The Question of Finiteness of the Labeling Algorithm; 6.4 Dijkstra's Algorithm; 6.5 The Floyd-Warshall Algorithm

Chap. 7 Primal-Dual Algorithms for Min-Cost Flow 7.1 The Min-Cost Flow Problem; 7.2 Combinatorializing the Capacities--Algorithm Cycle; 7.3 Combinatorializing the Cost--Algorithm Buildup; 7.4 An Explicit Primal-Dual Algorithm for the Hitchcock Problem--Algorithm Alphabeta; 7.5 A Transformation of Min-Cost Flow to Hitchcock; 7.6 Conclusion

Chap. 8 Algorithms and Complexity 8.1 Computability; 8.2 Time Bounds; 8.3 The Size of an Instance; 8.4 Analysis of Algorithms; 8.5 Polynomial-Time Algorithms; 8.6 Simplex Is Not a Polynomial-Time Algorithm; 8.7 The Ellipsoid Algorithm; 8.7.1 LP, LI, and LSI; 8.7.2 Affine Transformations and Ellipsoids; 8.7.3 The Algorithm; 8.7.4 Arithmetic Precision

Chap. 9 Efficient Algorithms for the Max-Flow Problem 9.1 Graph Search; 9.2 What Is Wrong With the Labeling Algorithm; 9.3 Network Labeling and Digraph Search; 9.4 An O(|V|²) Max-Flow Algorithm; 9.5 The Case of Unit Capacities

Chap. 10 Algorithms For Matching 10.1 The Matching Problem; 10.2 A Bipartite Matching Algorithm; 10.3 Bipartite Matching and Network Flow; 10.4 Nonbipartite Matching: Blossoms; 10.5 Nonbipartite Matching: An Algorithm

Chap. 11 Weighted Matching 11.1 Introduction; 11.2 The Hungarian Method for the Assignment Problem; 11.3 The Nonbipartite Weighted Matching Problem; 11.4 Conclusions

Chap. 12 Spanning Trees and Matroids 12.1 The Minimum Spanning Tree Problem; 12.2 An O(|E|log|V|) Algorithm for the Minimum Spanning Tree Problem; 12.3 The Greedy Algorithm; 12.4 Matroids; 12.5 The Intersection of Two Matroids; 12.6 On Certain Extensions of the Matroid Intersection Problem; 12.6.1 Weighted Matroid Intersection; 12.6.2 Matroid Parity; 12.6.3 The Intersection of Three Matroids

Chap. 13 Interger Linear Programming 13.1 Introduction; 13.2 Total Unimodularity; 13.3 Upper Bounds for Solutions of ILPs

Chap. 14 A Cutting-Plane Algorithm for Integer Linear Programs 14.1 Gomory Cuts; 14.2 Lexicography; 14.3 Finiteness of the Fractional Dual Algorithm; 14.4 Other Cutting-Plane Algorithms

Chap. 15 NP-Complete Problems 15.1 Introduction; 15.2 An Optimization Problem Is Three Problems; 15.3 The Classes P and NP; 15.4 Polynomial-Time Reductions; 15.5 Cook's Theorem; 15.6 Some Other NP-Complete Problems: Clique and the TSP; 15.7 More NP-Complete Problems: Matching, Covering, and Partitioning

Chap. 16 More About NP-Completeness 16.1 The Class co-NP; 16.2 Pseudo-Polynomial Algorithms and "Strong" NP-Complete Problems; 16.3 Special Cases and Generalizations of NP-Complete Problems; 16.3.1 NP-Completeness By Restriction; 16.3.2 Easy Special Cases of NP-Complete Problems; 16.3.3 Hard Special Cases of NP-Complete Problems; 16.4 A Glossary of Related Concepts; 16.4.1 Polynomial-Time Reductions; 16.4.2 NP-Hard problems; 16.4.3 Nondeterministic Turing Machines; 16.4.4 Polynomial-Space Complete Problems; 16.5 Epilogue

Chap. 17 Approximation Algorithms 17.1 Heuristics for Node Cover: An Example; 17.2 Approximation Algorithm for the Traveling Salesman Problem; 17.3 Approximation Schemes; 17.4 Negative Results

Chap. 18 Branch-and-Bound and Dynamic Programming 18.1 Branch-and-Bound for Integer Linear Programming; 18.2 Branch-and-Bound in a General Context; 18.3 Dominance Relations; 18.4 Branch-and-Bound Strategies; 18.5 Application to a Flowshop Scheduling Problem; 18.6 Dynamic Programming

Chap. 19 Local Search 19.1 Introduction; 19.2 Problem 1: The TSP; 19.3 Problem 2: Minimum-Cost Survivable Networks; 19.4 Problem 3: Topology of Offshore Natural Gas Pipeline Systems; 19.5 Problem 4: Uniform Graph Partitioning; 19.6 General Issues in Local Search; 19.7 The Geometry of Local Search; 19.8 An Example of a Large Minimal Exact Neighborhood; 19.9 The Complexity of Exact Local Search for the TSP

All chapters have problem sets and notes and references.

As can be seen, this book has a mighty amount of information, and it is amazingly well-explained. Of course, you need a firm grasp of your linear algebra, and some knowledge of very elementary calc./real analysis and graph theory (although most of the graph theory needed, technically speaking, is supplied in an appendix). You don't even really need to know a programming language, since the authors use a "pidgin algol," explained in yet another appendix, for most of the algorithm stuff; all it takes is an orderly thought process to follow it.

Despite the book's age, it mostly holds up very well in terms of topics and presentation. In the preface to the Dover edition, the authors briefly discuss some more current topics not dealt with in the text and make some (probably also out of date!) referrals for those wishing to "catch up." All in all, this book is a great value both as a text and a reference.

As a computer science graduate student I carried Papadimitriou and Steiglitz with me almost every day. Its target subject is combinatorial optimization, but going through this book, you might think that graph theory and computational complexity are just subfields of combinatorial optimization. It builds a beautiful theory that brings these and other fields together, and with a fraction of the page count of, say, Cormen, Rivest Leiserson. Now that it's a Dover book, it's a fraction of the price I paid, and I was gladly willing to pay that.

It is my favorite book on combinatorial optimization. The last 5 chapters 15-19 are the most interesting and useful to me because my job is write heuristics for NP hard problems in transportation. Chatpers 15 and 16 on NP complete problems are well explained and covered in depth. Chapter 17 on approximation algorithms is easy to understand and fun to read. Chapters 18 (branch-and-bound and dynamic programming) and 19 (local search) are very practical stuff, which I read many times.

The rest of the book is a good reference for topics like linear programming, max-flow, matching, etc. There are mostly independent of the last 5 chapters and can be skipped on a first read. My experience is that I don't need detailed knowledge of simplex algorithms because I use CPlex.

A couple of years ago I used this book to prepare for my PhD comprehensive exam and recently I picked it up again. While a lot has happened in OR since the book was written, the basics are covered in fine detail and in beautiful style. It could also be used as a graduate textbook or a course supplement. A very useful book for anyone interested in combinatorial optimization. What missing is the wealth of Artificial Intelligence techniques successfully used to tackle NP-hard optimization problems. But this is not a negative comment, just for your information. That belongs to another book. Linear Programming, Duality, Spanning Trees, Flows, Matching and introduction into NP-completeness are finely covered. A truly great job for an amazingly low price (see the text book prices with similar titles).

Effortless five stars. This is now almost 30 years old, and it is still a standard reference for the sorts of problems it addresses, in spite of the fact that those problems are mathematically interesting and often economically important, And of the standard references it is easily the cheapest, at least of those you have to pay anything at all for.

My only caveat is that I personally would have appreciated a more geometric tilt, rather than the algebraic one that the authors favour. Results tend to be derived first and primarily using algebra, and only then discussed from a geometric point of view. In particular, e.g., the discussion of duality is heavily algebraic, with Farkas' lemma tacked onto the end, _and proved using the previous algebraic result_. This seemed perverse to me, but your mileage may differ.

You could profitably also look at Boyd and Vandenberghe as well (speaking of things you don't _have_ to pay for).

I won't lie to you: this book is well written but relatively hard to read. The subject is inherently difficult, after all! I highly suggest it, though, because the author is a recognized expert on the field and the price is relatively low. It's worth it even if you enjoy a few pages...