Pseudorandom Self-Reductions for NP-Complete Problems

11 A language L is random-self-reducible if deciding membership in L can be reduced (in polynomial 12 time) to deciding membership in L for uniformly random instances. It is known that several “number 13 theoretic” languages (such as computing the permanent of a matrix) admit random self-reductions. 14 Feigenbaum and Fortnow showed that NP-complete languages are not non-adaptively random-self15 reducible unless the polynomial-time hierarchy collapses, giving suggestive evidence that NP may 16 not admit random self-reductions. Hirahara and Santhanam introduced a weakening of random 17 self-reductions that they called pseudorandom self-reductions, in which a language L is reduced to 18 a distribution that is computationally indistinguishable from the uniform distribution. They then 19 showed that the Minimum Circuit Size Problem (MCSP) admits a non-adaptive pseudorandom 20 self-reduction, and suggested that this gave further evidence that distinguished MCSP from standard 21 NP-Complete problems. 22 We show that, in fact, the Clique problem admits a non-adaptive pseudorandom self-reduction, 23 assuming the planted clique conjecture. More generally we show the following. Call a property of 24 graphs π hereditary if G ∈ π implies H ∈ π for every induced subgraph of G. We show that for any 25 infinite hereditary property π, the problem of finding a maximum induced subgraph H ∈ π of a 26 given graph G admits a non-adaptive pseudorandom self-reduction. 27 2012 ACM Subject Classification Theory of computation → Problems, reductions and completeness 28