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DOI: 10.4230/LIPIcs.ITCS.2025.54

Partial Minimum Branching Program Size Problem Is ETH-Hard

Authors: Ludmila Glinskih and Artur Riazanov

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

We show that assuming the Exponential Time Hypothesis, the Partial Minimum Branching Program Size Problem ({MBPSP}^{*}) requires superpolynomial time. This result also applies to the partial minimization problems for many interesting subclasses of branching programs, such as read-k branching programs and OBDDs. Combining these results with the recent unconditional lower bounds for {MCSP} [Ludmila Glinskih and Artur Riazanov, 2022], we obtain an unconditional superpolynomial lower bound on the size of Read-Once Nondeterministic Branching Programs (1- NBP) computing the total versions of the minimum BP, read-k-BP, and OBDD size problems. Additionally we show that it is NP-hard to check whether a given BP computing a partial Boolean function can be compressed to a BP of a given size.

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Ludmila Glinskih and Artur Riazanov. Partial Minimum Branching Program Size Problem Is ETH-Hard. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 54:1-54:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{glinskih_et_al:LIPIcs.ITCS.2025.54,
  author =	{Glinskih, Ludmila and Riazanov, Artur},
  title =	{{Partial Minimum Branching Program Size Problem Is ETH-Hard}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{54:1--54:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.54},
  URN =		{urn:nbn:de:0030-drops-226822},
  doi =		{10.4230/LIPIcs.ITCS.2025.54},
  annote =	{Keywords: MCSP, branching programs, meta-complexity, lower bounds}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2017.26

Satisfiable Tseitin Formulas Are Hard for Nondeterministic Read-Once Branching Programs

Authors: Ludmila Glinskih and Dmitry Itsykson

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract

We consider satisfiable Tseitin formulas TS_{G,c} based on d-regular expanders G with the absolute value of the second largest eigenvalue less than d/3. We prove that any nondeterministic read-once branching program (1-NBP) representing TS_{G,c} has size 2^{\Omega(n)}, where n is the number of vertices in G. It extends the recent result by Itsykson at el. [STACS 2017] from OBDD to 1-NBP. On the other hand it is easy to see that TS_{G,c} can be represented as a read-2 branching program (2-BP) of size O(n), as the negation of a nondeterministic read-once branching program (1-coNBP) of size O(n) and as a CNF formula of size O(n). Thus TS_{G,c} gives the best possible separations (up to a constant in the exponent) between 1-NBP and 2-BP, 1-NBP and 1-coNBP and between 1-NBP and CNF.

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Ludmila Glinskih and Dmitry Itsykson. Satisfiable Tseitin Formulas Are Hard for Nondeterministic Read-Once Branching Programs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 26:1-26:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{glinskih_et_al:LIPIcs.MFCS.2017.26,
  author =	{Glinskih, Ludmila and Itsykson, Dmitry},
  title =	{{Satisfiable Tseitin Formulas Are Hard for Nondeterministic Read-Once Branching Programs}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{26:1--26:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.26},
  URN =		{urn:nbn:de:0030-drops-80767},
  doi =		{10.4230/LIPIcs.MFCS.2017.26},
  annote =	{Keywords: Tseitin formula, read-once branching program, expander}
}

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Documents authored by Glinskih, Ludmila

Partial Minimum Branching Program Size Problem Is ETH-Hard

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Satisfiable Tseitin Formulas Are Hard for Nondeterministic Read-Once Branching Programs

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