Minimum Circuit Size, Graph Isomorphism, and Related Problems

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. * This work subsumes and significantly strengthens the earlier arXiv submission 1511.08189 [AGM15]. See the end of Section 1 for more details.

Conhecimento Zero Estatístico e Reduções Eficientes para o Problema MKTP

“…Também obtivemos resultados relacionando os problemas HSP e MKTP. Mostramos que HSP ∈ BPP MKTP adaptando um resultado de [Allender et al 2018]. Além disso, mostramos queé possível tornar essa redução mais forte quando há uma maneira eficiente de estimar a ordem do grupo dado como entrada para o problema, implicando que, nesses casos, HSP ∈ ZPP MKTP .…”

“…Estudamos ainda uma técnica bastante recente de redução para MKTP [Allender et al 2018]. Essa técnica consiste no uso do oráculo para MKTP como um estimador da entropia (ver [Sdroievski 2018, Seção 4.3 Além disso, essas técnicas permitem mostrar reduções mais fortes para uma classe de problemas em grupos que inclui os problemas ISOMORFISMO DE GRAFOS e RESÍDUO QUADRÁTICO [Allender et al 2018].…”

Vaughan Jones, Kolmogorov Complexity, and the New Complexity Landscape around Circuit Minimization

“…Furthermore it has frequently been the case that a theorem about MCSP was first proved for MKTP and then the result for MCSP was obtained as a corollary. However, there is no efficient reduction known (in either direction) between MCSP and MKTP, and there are some theorems that are currently known to hold only for MKTP, although they are suspected to hold also for MCSP (e.g., [8,10,31,20]). 3 Similarly, some of the more intriguing recent developments can only be understood by paying attention to the distinction between different notions of resource-bounded Kolmogorov complexity.…”

The New Complexity Landscape Around Circuit Minimization

“…Furthermore it has frequently been the case that a theorem about MCSP was first proved for MKTP and then the result for MCSP was obtained as a corollary. However, there is no efficient reduction known (in either direction) between MCSP and MKTP, and there are some theorems that are currently known to hold only for MKTP, although they are suspected to hold also for MCSP (e.g., [4,6,23]). Similarly, some of the more intriguing recent developments can only be understood by paying attention to the distinction between different notions of resource-bounded Kolmogorov complexity.…”

“…SZK also contains the Graph Isomorphism problem, which is ≤ RP T -reducible to MCSP and MKTP. In [4], Graph Isomorphism (and several other problems) were shown to be ≤ ZPP T reducible to MKTP; it remains unknown if this also holds for MCSP. In fact, there is no interesting example of a problem A that is not known to be in NP ∩ coNP that has been shown to be ≤ ZPP T reducible to MCSP.…”