The problem of determining whether two given finite graphs are isomorphic is easily seen to belong to the class NP. But up to now, no polynomial time algorithm is known. On the other hand, no NP-completeness proof is known either. That is, GRAPH ISOMORPHISM is one of the few “open problems” in NP according to Garey and Johnson’s terminology [9]. Note that GRAPH ISOMORPHISM is already mentioned ...

We consider the analog of the P versus NP∩ co-NP question for the classical two-party communication protocols where polynomial time is replaced by poly-logarithmic communication: if both a boolean function f and its negation ¬f have small (poly-logarithmic in the number of variables) nondeterministic communication complexity, what is then its deterministic and/or probabilistic communication com...

The Valiant-Vazirani Isolation Lemma [TCS, vol. 47, pp. 85–93, 1986] provides an efficient procedure for isolating a satisfying assignment of a given satisfiable circuit: given a Boolean circuit C on n input variables, the procedure outputs a new circuit C ′ on the same n input variables with the property that the set of satisfying assignments for C ′ is a subset of those for C, and moreover, i...

We show that if an NP-complete problem has a non-adaptive self-corrector with respect to a samplable distribution then coNP is contained in NP/poly and the polynomial hierarchy collapses to the third level. Feigenbaum and Fortnow (SICOMP 22:994-1005, 1993) show the same conclusion under the stronger assumption that an NP-complete problem has a non-adaptive random self-reduction. Our result show...