# Graph theory

## Minimum Circuit Size Problem and Graph Isomorphism solved

The Saga of NP-Intermediate Problems: A New Development in an Old Story Eric Allender

Eric Allender

Rutgers University

Abstract: For roughly four decades, two of the best-studied problems in NP that are not known to be in P or to be NP complete have been:

- Graph Isomorphism, and

- MCSP (the Minimum Circuit Size Problem).

Yet there had been no theorem, relating the complexity of these two problems to each other.

Until now. We give a simple argument — drawing on the connection between MCSP and time-bounded Kolmogorov complexity — showing that not only Graph Isomorphism, but every problem in the complexity class SZK (Statistical Zero Knowledge) is BPP reducible to MCSP.

Joint work with Bireswar Das: http://ftp.cs.rutgers.edu/pub/allender/szk.mcsp.pdf.

SFI Host: Cris Moore

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