Graph Problems: Hard Problems
 Steven S. Skiena
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Abstract
A cynical view of graph algorithms is that “everything we want to do is hard.”Indeed, no polynomialtime algorithms are known for any of the problems in this section.All of them are provably NPcomplete with the exception of graph isomorphism—whose complexity status remains an open question.
The theory of NPcompleteness demonstrates that all NPcomplete problems must have polynomialtime algorithms if any one of them does. This prospect is sufficiently preposterous that an NPcompleteness reduction suffices as de facto proof that no efficient algorithm exists to solve the given problem.
Still, do not abandon hope if your problem resides in this chapter. We provide a recommended line of attack for each problem, be it combinatorial search, heuristics, approximation algorithms, or algorithms for restricted instances. Hard problems require a different methodology to work with than polynomialtime problems, but with care they can usually be dealt with successfully.
 M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the theory of NPcompleteness. W. H. Freeman, San Francisco, 1979.
 T. Gonzalez. Handbook of Approximation Algorithms and Metaheuristics. ChapmanHall / CRC, 2007.
 D. Hochbaum, editor. Approximation Algorithms for NPhard Problems. PWS Publishing, Boston, 1996.
 V. Vazirani. Approximation Algorithms. Springer, 2004.
 Title
 Graph Problems: Hard Problems
 Book Title
 The Algorithm Design Manual
 Book Part
 Part II
 Pages
 pp 523561
 Copyright
 2008
 DOI
 10.1007/9781848000704_16
 Print ISBN
 9781848000698
 Online ISBN
 9781848000704
 Publisher
 Springer London
 Copyright Holder
 SpringerVerlag London Limited
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 Authors

 Steven S. Skiena ^{(2)}
 Author Affiliations

 2. Department of Computer Science, State University of New York at Stony Brook, New York, USA
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