TY - JOUR
TI - Minimum Circuit Size, Graph Isomorphism, and Related Problems
DO - https://doi.org/doi:10.7282/T3HH6PBZ
AU - Allender, Eric
AU - Grochow, Joshua A.
AU - van Melkebeek, Dieter
AU - Moore, Cristopher
AU - Morgan, Andrew
PY - 2018
T2 - SIAM Journal on Computing
AB - 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 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 satises 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.
KW - Reductions between NP-intermediate problems
KW - Graph Isomorphism
KW - Minimum Circuit Size Problem
KW - Time-bounded Kolmogorov complexity
LA - English
ER -