Minimum Circuit Size, Graph Isomorphism, and Related Problems

TL;DR

This paper introduces a novel randomized reduction from Graph Isomorphism and related problems to the Minimum Kolmogorov Time Problem, using encodings that are efficiently decodable and near optimal in compression.

Contribution

It presents a new approach to relate isomorphism problems to MCSP/MKTP via interactive proof-inspired reductions, expanding the scope beyond previous methods.

Findings

Established a zero-sided error randomized reduction from GI to MKTP.

Generalized the reduction to other isomorphism problems with specific group properties.

Developed efficient, near-optimal encodings of isomorphism classes.

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. All prior 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.