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The graph isomorphism problem: its structural complexityAugust 1994
- Authors:
- Johannes Köbler,
- Uwe Schöning,
- Jacobo Torán
Abstract
No abstract available.
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Index Terms
- The graph isomorphism problem: its structural complexity
Reviews
Douglas M. Campbell
The authors summarize recent results on the complexity of the graph isomorphism (GI) problem relative to NP and other complexity classes, and explain the aims and scope of structural complexity theory. Presuming only an elementary knowledge of complexity theory and probability theory, the book contains wonderful examples for a standard complexity theory course and makes its results accessible to a broad audience. The book is divided into four chapters. The first chapter develops reducibility, NP-completeness, and the difference between deciding a problem, solving a problem, and counting the number of solutions to a problem. The second chapter deals with the polynomial hierarchy, interactive proofs, probabilistic complexity classes, and the concept of “lowness” relative to GI. In particular, if the polynomial hierarchy does not collapse to S p 2 , then GI is not NP-complete. The third chapter studies reducing intractable problems to sets with low information content, such as sparse sets. Although it is unknown whether an NP-complete problem can be sparse, one consequence of this section is that GI is “almost” in NP ? co-NP . Finally, chapter 4 deals with counting the number of solutions to a problem. Unlike some other problems, counting the number of isomorphisms for GI is not harder than deciding whether an isomorphism exists for GI. The chapter ends by relating Ulam's classical graph re<__?__Pub Caret>construction problem to GI. The book is loaded with examples, is pleasant reading, and is excellent for a graduate seminar.
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