Complexity classes of equivalence problems revisited

To determine if two lists of numbers are the same set, we sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms arise in graph isomorphism algorithms. To determine if two graphs are cospectral (have...

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Bibliographic Details
Published in:Information and computation Vol. 209; no. 4; pp. 748 - 763
Main Authors: Fortnow, Lance, Grochow, Joshua A.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier Inc 01.04.2011
Elsevier
Subjects:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Canonical form
Complexity class
Computational complexity
Computer science; control theory; systems
Equivalence relation
Exact sciences and technology
Information retrieval. Graph
Isomorphism problem
Mathematics
Miscellaneous
Normal form
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Oracle
Probabilistic computation
Quantum computation
Sciences and techniques of general use
Theoretical computing
Probabilistic computation
Equivalence relation
Complexity class
Quantum computation
Normal form
Isomorphism problem
Oracle
Computational complexity
Canonical form
Invariant
Computer theory
Characteristic polynomial
Eigenvalue
Equivalence classes
Graph algorithm
ISSN:0890-5401, 1090-2651
Online Access:Get full text
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Summary:To determine if two lists of numbers are the same set, we sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms arise in graph isomorphism algorithms. To determine if two graphs are cospectral (have the same eigenvalues), we compute their characteristic polynomials and see if they are equal; the characteristic polynomial is a complete invariant for cospectrality. Finally, an equivalence relation may be decidable in P without either a complete invariant or canonical form. Blass and Gurevich (1984) asked whether these conditions on equivalence relations—having an FP canonical form, having an FP complete invariant, and being in P—are distinct. They showed that this question requires non-relativizing techniques to resolve. We extend their results, and give new connections to probabilistic and quantum computation.
ISSN:0890-5401
1090-2651
DOI:10.1016/j.ic.2011.01.006