NDLI: Minimum Circuit Size, Graph Isomorphism, and Related Problems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Allender, Eric van Melkebeek, Dieter Moore, Cristopher Morgan, Andrew Grochow, Joshua A.
Copyright Year	2018
Abstract	We study the computational power of deciding whether a given truth table can be described by a circuit of a given size (the minimum circuit size problem, or MCSP for short) and of the variant denoted as MKTP, where circuit size is replaced by a polynomially related Kolmogorov measure. Prior to our work, all reductions from supposedly intractable problems to MCSP/MKTP hinged on the power of MCSP/MKTP to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive proof system for the complement of graph isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include linear code equivalence, permutation group conjugacy, and matrix subspace conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest.
Sponsorship	Santa Fe Institute. National Science Foundation
Starting Page	1339
Ending Page	1372
Page Count	34
File Format	PDF
ISSN	00975397
DOI	10.1137/17M1157970
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	47
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-07-12
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems time-bounded Kolmogorov complexity minimum circuit size problem graph isomorphism reductions between NP-intermediate problems Complexity classes Algorithmic information theory
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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