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Pseudorandom Self-Reductions for NP-Complete Problems

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

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

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.

Cite as

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)

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  author =	{Elrazik, Reyad Abed and Robere, Robert and Schuster, Assaf and Yehuda, Gal},
  title =	{{Pseudorandom Self-Reductions for NP-Complete Problems}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{65:1--65:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.65},
  URN =		{urn:nbn:de:0030-drops-156615},
  doi =		{10.4230/LIPIcs.ITCS.2022.65},
  annote =	{Keywords: computational complexity, pseudorandomness, worst-case to average-case, self reductions, planted clique, hereditary graph family}
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