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Total Functions in the Polynomial Hierarchy

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

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Author Details

Robert Kleinberg
  • Cornell University, Ithaca, NY, USA
Oliver Korten
  • Columbia University, New York, NY, USA
Daniel Mitropolsky
  • Columbia University, New York, NY, USA
Christos Papadimitriou
  • Columbia University, New York, NY, USA

Funding

  • Mitropolsky, Daniel: This research was supported in part by a grant from the Columbia-IBM center for Blockchain and Data Transparency, and by JPMorgan Chase & Co. This research was supported in part by NSF Awards CCF1763970 andCCF1910700, and by a research contract with Softbank
  • Papadimitriou, Christos: This research was supported in part by NSF Awards CCF1763970 andCCF1910700, and by a research contract with Softbank.

Acknowledgements

Many thanks to Noga Alon for a very enlightening conversation about the union bound, and to Omri Weinstein for an interesting discussion. The authors also thank Ofer Grossman and Eylon Yogev for useful discussions after the pre-print.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ITCS.2021.44

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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
Keywords
  • total complexity
  • polynomial hierarchy
  • pigeonhole principle

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