CNF and DNF succinct graph encodings

Abstract

It is well-known that succinct encodings of computational problems – using circuits or formulas to encode large instances – generally result in an exponential complexity blow-up compared to their original complexity.

We introduce a new way to encode graph problems, based on CNF or DNF formulas. We show that – contrary to the other existing succinct models – there are examples of problems whose complexity does not increase when encoded in the new form, or increases to an intermediate complexity class less powerful than the exponential blow up.

We also study the complexity of the succinct versions of the Graph Isomorphism problem. We show that all the versions are hard for PSPACE. Although the exact complexity of these problems is still unknown, we show that under most existing succinct models the different versions of the problem are equivalent. We also give an algorithm for the DNF encoded version of GI whose running time depends mainly on the number of terms in the succinct representation.