## Graph Nonisomorphism Has Subexponential Size Proofs Unless The Polynomial-Time Hierarchy Collapses (1998)

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Venue: | SIAM Journal on Computing |

Citations: | 108 - 6 self |

### BibTeX

@ARTICLE{Klivans98graphnonisomorphism,

author = {Adam R. Klivans and Dieter Van Melkebeek},

title = {Graph Nonisomorphism Has Subexponential Size Proofs Unless The Polynomial-Time Hierarchy Collapses},

journal = {SIAM Journal on Computing},

year = {1998},

volume = {31},

pages = {659--667}

}

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### Abstract

We establish hardness versus randomness trade-offs for a broad class of randomized procedures. In particular, we create efficient nondeterministic simulations of bounded round Arthur-Merlin games using a language in exponential time that cannot be decided by polynomial size oracle circuits with access to satisfiability. We show that every language with a bounded round Arthur-Merlin game has subexponential size membership proofs for infinitely many input lengths unless exponential time coincides with the third level of the polynomial-time hierarchy (and hence the polynomial-time hierarchy collapses). This provides the first strong evidence that graph nonisomorphism has subexponential size proofs. We set up a general framework for derandomization which encompasses more than the traditional model of randomized computation. For a randomized procedure to fit within this framework, we only require that for any fixed input the complexity of checking whether the procedure succeeds on a given ...