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Pseudorandom Self-Reductions for NP-Complete Problems

Authors Reyad Abed Elrazik, Robert Robere, Assaf Schuster, Gal Yehuda



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Author Details

Reyad Abed Elrazik
  • Taub Faculty of Computer Science, Technion, Haifa, Israel
Robert Robere
  • School of Computer Science, McGill University, Montreal, Canada
Assaf Schuster
  • Taub Faculty of Computer Science, Technion, Haifa, Israel
Gal Yehuda
  • Taub Faculty of Computer Science, Technion, Haifa, Israel

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Reyad Abed Elrazik, Robert Robere, Assaf Schuster, and Gal Yehuda. Pseudorandom Self-Reductions for NP-Complete Problems. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 65:1-65:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.65

Abstract

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 distinguished MCSP from standard NP-Complete problems. We show that, in fact, the 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 π hereditary if G ∈ π implies H ∈ π for every induced subgraph of G. We show that for any infinite hereditary property π, the problem of finding a maximum induced subgraph H ∈ π of a given graph G admits a non-adaptive pseudorandom self-reduction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • computational complexity
  • pseudorandomness
  • worst-case to average-case
  • self reductions
  • planted clique
  • hereditary graph family

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