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DOI: 10.4230/LIPIcs.ITCS.2023.67

Downward Self-Reducibility in TFNP

Authors: Prahladh Harsha, Daniel Mitropolsky, and Alon Rosen

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract

A problem is downward self-reducible if it can be solved efficiently given an oracle that returns solutions for strictly smaller instances. In the decisional landscape, downward self-reducibility is well studied and it is known that all downward self-reducible problems are in PSPACE. In this paper, we initiate the study of downward self-reducible search problems which are guaranteed to have a solution - that is, the downward self-reducible problems in TFNP. We show that most natural PLS-complete problems are downward self-reducible and any downward self-reducible problem in TFNP is contained in PLS. Furthermore, if the downward self-reducible problem is in TFUP (i.e. it has a unique solution), then it is actually contained in UEOPL, a subclass of CLS. This implies that if integer factoring is downward self-reducible then it is in fact in UEOPL, suggesting that no efficient factoring algorithm exists using the factorization of smaller numbers.

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Prahladh Harsha, Daniel Mitropolsky, and Alon Rosen. Downward Self-Reducibility in TFNP. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 67:1-67:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{harsha_et_al:LIPIcs.ITCS.2023.67,
  author =	{Harsha, Prahladh and Mitropolsky, Daniel and Rosen, Alon},
  title =	{{Downward Self-Reducibility in TFNP}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{67:1--67:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.67},
  URN =		{urn:nbn:de:0030-drops-175700},
  doi =		{10.4230/LIPIcs.ITCS.2023.67},
  annote =	{Keywords: downward self-reducibility, TFNP, TFUP, factoring, PLS, CLS}
}

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DOI: 10.4230/LIPIcs.ITCS.2021.44

Total Functions in the Polynomial Hierarchy

Authors: Robert Kleinberg, Oliver Korten, Daniel Mitropolsky, and Christos Papadimitriou

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract

We identify several genres of search problems beyond NP for which existence of solutions is guaranteed. One class that seems especially rich in such problems is PEPP (for "polynomial empty pigeonhole principle"), which includes problems related to existence theorems proved through the union bound, such as finding a bit string that is far from all codewords, finding an explicit rigid matrix, as well as a problem we call Complexity, capturing Complexity Theory’s quest. When the union bound is generous, in that solutions constitute at least a polynomial fraction of the domain, we have a family of seemingly weaker classes α-PEPP, which are inside FP^NP|poly. Higher in the hierarchy, we identify the constructive version of the Sauer-Shelah lemma and the appropriate generalization of PPP that contains it, as well as the problem of finding a king in a tournament (a vertex k such that all other vertices are defeated by k, or by somebody k defeated).

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Robert Kleinberg, Oliver Korten, Daniel Mitropolsky, and Christos Papadimitriou. Total Functions in the Polynomial Hierarchy. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kleinberg_et_al:LIPIcs.ITCS.2021.44,
  author =	{Kleinberg, Robert and Korten, Oliver and Mitropolsky, Daniel and Papadimitriou, Christos},
  title =	{{Total Functions in the Polynomial Hierarchy}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{44:1--44:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.44},
  URN =		{urn:nbn:de:0030-drops-135835},
  doi =		{10.4230/LIPIcs.ITCS.2021.44},
  annote =	{Keywords: total complexity, polynomial hierarchy, pigeonhole principle}
}

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Downward Self-Reducibility in TFNP

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Total Functions in the Polynomial Hierarchy

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