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### Paper:

TR21-170 | 25th November 2021 20:35

#### Pseudorandom Self-Reductions for NP-Complete Problems

TR21-170
Authors: Reyad Abed Elrazik, Robert Robere, Assaf Schuster, Gal Yehuda
Publication: 28th November 2021 12:18
A language $L$ is random-self-reducible if deciding membership in $L$ can be reduced (in polynomial time) to deciding membership in $L$ for uniformly random instances. It is known that several "number theoretic" languages (such as computing the permanent of a matrix) admit random self-reductions. Feigenbaum and Fortnow showed that NP-complete languages are not non-adaptively random-self-reducible unless the polynomial-time hierarchy collapses, giving suggestive evidence that NP may not admit random self-reductions. Hirahara and Santhanam introduced a weakening of random self-reductions that they called pseudorandom self-reductions, in which a language $L$ is reduced to a distribution that is computationally indistinguishable from the uniform distribution. They then showed that the Minimum Circuit Size Problem (MCSP) admits a non-adaptive pseudorandom self-reduction, and suggested that this gave further evidence that MCSP is "distinguished" from standard NP-Complete problems.
We show that, in fact, the NP-Complete Clique problem admits a non-adaptive pseudorandom self-reduction, assuming the planted clique conjecture. More generally we show the following. Call a property of graphs $\pi$ hereditary if $G \in \pi$ implies $H \in \pi$ for every induced subgraph of $G$. We show that for any infinite hereditary property $\pi$, the problem of finding a maximum induced subgraph $H \in \pi$ of a given graph $G$ admits a non-adaptive pseudorandom self-reduction.