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DOI: 10.4230/LIPIcs.FSTTCS.2025.47

The Algebraic Cost of a Boolean Sum

Authors: Ian Orzel, Srikanth Srinivasan, Sébastien Tavenas, and Amir Yehudayoff

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract

It is a well-known fact that the permanent polynomial is complete for the complexity class VNP, and it is largely suspected that the determinant does not share this property, despite its similar expression. We study the question of why the VNP-completeness proof of the permanent fails for the determinant. We isolate three fundamental properties that are sufficient to prove a polynomial sequence is VNP-hard, of which two are shared by both the permanent and the determinant. We proceed to show that the permanent satisfies the third property, which we refer to as the "cost of a boolean sum", while the determinant does not, showcasing the fundamental difference between the polynomial families. We further note that this differentiation also applies in the border complexity setting and that our results apply for counting complexity.

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Ian Orzel, Srikanth Srinivasan, Sébastien Tavenas, and Amir Yehudayoff. The Algebraic Cost of a Boolean Sum. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 47:1-47:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{orzel_et_al:LIPIcs.FSTTCS.2025.47,
  author =	{Orzel, Ian and Srinivasan, Srikanth and Tavenas, S\'{e}bastien and Yehudayoff, Amir},
  title =	{{The Algebraic Cost of a Boolean Sum}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{47:1--47:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.47},
  URN =		{urn:nbn:de:0030-drops-251271},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.47},
  annote =	{Keywords: Algebraic Complexity, Computational Complexity, Permanent, Determinant}
}

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The Algebraic Cost of a Boolean Sum

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