The graph isomorphism problem : its structural complexity
著者
書誌事項
The graph isomorphism problem : its structural complexity
（Progress in theoretical computer science）
Birkhäuser, c1993
- : us
- : gw
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注記
Includes bibliographical references (p. [149]-157) and index
内容説明・目次
- 巻冊次
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: us ISBN 9780817636807
内容説明
Recently, a variety ofresults on the complexitystatusofthegraph isomorphism problem has been obtained. These results belong to the so-called structural part of Complexity Theory. Our idea behind this book is to summarize such results which might otherwise not be easily accessible in the literature, and also, to give the reader an understanding of the aims and topics in Structural Complexity Theory, in general. The text is basically self contained; the only prerequisite for reading it is some elementary knowledge from Complexity Theory and Probability Theory. It can be used to teach a seminar or a monographic graduate course, but also parts of it (especially Chapter 1) provide a source of examples for a standard graduate course on Complexity Theory. Many people have helped us in different ways III the process of writing this book. Especially, we would like to thank V. Arvind, R.V. Book, E. May ordomo, and the referee who gave very constructive comments. This book project was especially made possible by a DAAD grant in the "Acciones In tegrada" program. The third author has been supported by the ESPRIT project ALCOM-II.
目次
Preliminaries.- 1 Decision Problems, Search Problems, and Counting Problems.- 1.1 NP-Completeness.- 1.1.1 The Classes P and NP.- 1.1.2 Reducibility.- 1.2 Reducing the Construction Problem to the Decision Problem.- 1.3 Counting versus Deciding for Graph Isomorphism.- 1.4 Uniqueness of the Solution.- 1.4.1 Random Reductions.- 1.4.2 Promise Problems.- 1.5 Reducing Multiple Questions to One.- 2 Quantifiers, Games, and Interactive Proofs.- 2.1 The Polynomial-Time Hierarchy.- 2.2 Interactive Proof Systems.- 2.2.1 The Class IP.- 2.2.2 Zero-Knowledge.- 2.3 Probabilistic Classes.- 2.3.1 Probability Amplification.- 2.3.2 The BP-Operator.- 2.3.3 Arthur-Merlin Games.- 2.4 Lowness and Collapses.- 3 Circuits and Sparse Sets.- 3.1 Polynomial Size Circuits.- 3.1.1 Circuits for NP.- 3.1.2 Circuits for Graph Isomorphism.- 3.2 Reductions to Sparse Sets.- 4 Counting Properties.- 4.1 Decision Reduces to Parity.- 4.2 Graph Isomorphism is Low for PP.- 4.3 The Reconstruction Conjecture.
- 巻冊次
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: gw ISBN 9783764336806
内容説明
The graph isomorphism problem belongs to the part of Complexity Theory that focuses on the structure of complexity classes involved in the classification of computational problems and in the relations among them. It consists in deciding whether two given graphs are isomorphic, i.e. whether there is a bijective mapping from the nodes of one graph to the nodes of the second graph such that the edge connections are respected. It is a problem of considerable practical as wen as theoretical importance that is, as of now, unresolved in the sense that no efficient algorithm for it has yet been found. Given this fact, it is natural to ask whether such an algorithm exists at an or whether the problem is intractable. -Be book focuses on this issue and presents several recent results that provide a better understanding of the relative position of the graph isomorphism problem in the class NP as well as in other complexity classes. It also uses the problem to illustrate important concepts in structural complexity, providing a look into the more general theory.
'The book is basically self-contained; the only prerequisite for reading it is some elementary knowledge from Complexity Theory and Probability Theory. Its level of presentation makes it eminently suitable for a seminar or graduate course devoted to the problem, or as a rich source of examples for a standard graduate course in Complexity Theory.
目次
Introduction Preliminaries Decision Problems, Search Problems, and Counting Problems NP-Completeness The Classes P and NP Reducibility. Reducing the Construction Problem to the Decision Problem Counting versus Deciding for Graph Isomorphism Uniqueness of the Solution Random Reductions Promise Problems Reducing Multiple Questions to One Quantifiers, Games, and Interactive Proofs The Polynomial-Time Hierarchy Interactive Proof Systems The Class IP Zero-Knowledge Probabilistic Classes Probability Amplification The BP-Operator Arthur-Merlin Games Lowness and Collapses Circuits and Sparse Sets Polynomial Size Circuits Circuits for NP Circuits for Graph Isomorphism Reductions to Sparse Sets Counting Properties Decision Reduces to Parity Graph Isomorphism is low for PP The Reconstruction Conjecture Bibliography Index
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