5 Search Results for "Jukna, Stasys"

Document
DOI: 10.4230/LIPIcs.CCC.2021.17
Branching Programs with Bounded Repetitions and Flow Formulas

Authors: Anastasia Sofronova and Dmitry Sokolov

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
Restricted branching programs capture various complexity measures like space in Turing machines or length of proofs in proof systems. In this paper, we focus on the application in the proof complexity that was discovered by Lovasz et al. [László Lovász et al., 1995] who showed the equivalence between regular Resolution and read-once branching programs for "unsatisfied clause search problem" (Search_φ). This connection is widely used, in particular, in the recent breakthrough result about the Clique problem in regular Resolution by Atserias et al. [Albert Atserias et al., 2018]. We study the branching programs with bounded repetitions, so-called (1,+k)-BPs (Sieling [Detlef Sieling, 1996]) in application to the Search_φ problem. On the one hand, it is a natural generalization of read-once branching programs. On the other hand, this model gives a powerful proof system that can efficiently certify the unsatisfiability of a wide class of formulas that is hard for Resolution (Knop [Alexander Knop, 2017]). We deal with Search_φ that is "relatively easy" compared to all known hard examples for the (1,+k)-BPs. We introduce the first technique for proving exponential lower bounds for the (1,+k)-BPs on Search_φ. To do it we combine a well-known technique for proving lower bounds on the size of branching programs [Detlef Sieling, 1996; Detlef Sieling and Ingo Wegener, 1994; Stasys Jukna and Alexander A. Razborov, 1998] with the modification of the "closure" technique [Michael Alekhnovich et al., 2004; Michael Alekhnovich and Alexander A. Razborov, 2003]. In contrast with most Resolution lower bounds, our technique uses not only "local" properties of the formula, but also a "global" structure. Our hard examples are based on the Flow formulas introduced in [Michael Alekhnovich and Alexander A. Razborov, 2003].
Cite as

Anastasia Sofronova and Dmitry Sokolov. Branching Programs with Bounded Repetitions and Flow Formulas. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 17:1-17:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{sofronova_et_al:LIPIcs.CCC.2021.17,
  author =	{Sofronova, Anastasia and Sokolov, Dmitry},
  title =	{{Branching Programs with Bounded Repetitions and Flow Formulas}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{17:1--17:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.17},
  URN =		{urn:nbn:de:0030-drops-142915},
  doi =		{10.4230/LIPIcs.CCC.2021.17},
  annote =	{Keywords: proof complexity, branching programs, bounded repetitions, lower bounds}
}
Document
Extended Abstract
DOI: 10.4230/LIPIcs.ITCS.2021.89
Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0 (Extended Abstract)

Authors: Yuval Filmus, Or Meir, and Avishay Tal

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

Abstract
Håstad showed that any De Morgan formula (composed of AND, OR and NOT gates) shrinks by a factor of O(p²) under a random restriction that leaves each variable alive independently with probability p [SICOMP, 1998]. Using this result, he gave an Ω̃(n³) formula size lower bound for the Andreev function, which, up to lower order improvements, remains the state-of-the-art lower bound for any explicit function. In this work, we extend the shrinkage result of Håstad to hold under a far wider family of random restrictions and their generalization - random projections. Based on our shrinkage results, we obtain an Ω̃(n³) formula size lower bound for an explicit function computed in AC⁰. This improves upon the best known formula size lower bounds for AC⁰, that were only quadratic prior to our work. In addition, we prove that the KRW conjecture [Karchmer et al., Computational Complexity 5(3/4), 1995] holds for inner functions for which the unweighted quantum adversary bound is tight. In particular, this holds for inner functions with a tight Khrapchenko bound. Our random projections are tailor-made to the function’s structure so that the function maintains structure even under projection - using such projections is necessary, as standard random restrictions simplify AC⁰ circuits. In contrast, we show that any De Morgan formula shrinks by a quadratic factor under our random projections, allowing us to prove the cubic lower bound. Our proof techniques build on the proof of Håstad for the simpler case of balanced formulas. This allows for a significantly simpler proof at the cost of slightly worse parameters. As such, when specialized to the case of p-random restrictions, our proof can be used as an exposition of Håstad’s result.
Cite as

Yuval Filmus, Or Meir, and Avishay Tal. Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0 (Extended Abstract). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 89:1-89:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{filmus_et_al:LIPIcs.ITCS.2021.89,
  author =	{Filmus, Yuval and Meir, Or and Tal, Avishay},
  title =	{{Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{89:1--89:7},
  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.2021.89},
  URN =		{urn:nbn:de:0030-drops-136281},
  doi =		{10.4230/LIPIcs.ITCS.2021.89},
  annote =	{Keywords: De Morgan formulas, KRW Conjecture, shrinkage, random restrictions, random projections, bounded depth circuits, constant depth circuits, formula complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.41
Lower Bounds for DeMorgan Circuits of Bounded Negation Width

Authors: Stasys Jukna and Andrzej Lingas

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract
We consider Boolean circuits over {or, and, neg} with negations applied only to input variables. To measure the "amount of negation" in such circuits, we introduce the concept of their "negation width". In particular, a circuit computing a monotone Boolean function f(x_1,...,x_n) has negation width w if no nonzero term produced (purely syntactically) by the circuit contains more than w distinct negated variables. Circuits of negation width w=0 are equivalent to monotone Boolean circuits, while those of negation width w=n have no restrictions. Our motivation is that already circuits of moderate negation width w=n^{epsilon} for an arbitrarily small constant epsilon>0 can be even exponentially stronger than monotone circuits. We show that the size of any circuit of negation width w computing f is roughly at least the minimum size of a monotone circuit computing f divided by K=min{w^m,m^w}, where m is the maximum length of a prime implicant of f. We also show that the depth of any circuit of negation width w computing f is roughly at least the minimum depth of a monotone circuit computing f minus log K. Finally, we show that formulas of bounded negation width can be balanced to achieve a logarithmic (in their size) depth without increasing their negation width.
Cite as

Stasys Jukna and Andrzej Lingas. Lower Bounds for DeMorgan Circuits of Bounded Negation Width. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{jukna_et_al:LIPIcs.STACS.2019.41,
  author =	{Jukna, Stasys and Lingas, Andrzej},
  title =	{{Lower Bounds for DeMorgan Circuits of Bounded Negation Width}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{41:1--41:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.41},
  URN =		{urn:nbn:de:0030-drops-102801},
  doi =		{10.4230/LIPIcs.STACS.2019.41},
  annote =	{Keywords: Boolean circuits, monotone circuits, lower bounds}
}
Document
DOI: 10.4230/DagSemProc.06111.8
Graphs and Circuits: Some Further Remarks

Authors: Stasys Jukna

Published in: Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

Abstract
We consider the power of single level circuits in the context of graph complexity. We first prove that the single level conjecture fails for fanin-$2$ circuits over the basis ${oplus,land,1}$. This shows that the (surpisingly tight) phenomenon, established by Mirwald and Schnorr (1992) for quadratic functions, has no analogon for graphs. We then show that the single level conjecture fails for unbounded fanin circuits over ${lor,land,1}$. This partially answers the question of Pudl'ak, R"odl and Savick'y (1986). We also prove that $Sigma_2 eq Pi_2$ in a restricted version of the hierarhy of communication complexity classes introduced by Babai, Frankl and Simon (1986). Further, we show that even depth-$2$ circuits are surprisingly powerful: every bipartite $n imes n$ graph of maximum degree $Delta$ can be represented by a monotone CNF with $O(Deltalog n)$ clauses. We also discuss a relation between graphs and $ACC$-circuits.
Cite as

Stasys Jukna. Graphs and Circuits: Some Further Remarks. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{jukna:DagSemProc.06111.8,
  author =	{Jukna, Stasys},
  title =	{{Graphs and Circuits: Some Further Remarks}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--16},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.8},
  URN =		{urn:nbn:de:0030-drops-6218},
  doi =		{10.4230/DagSemProc.06111.8},
  annote =	{Keywords: Graph complexity, single level conjecture, Sylvester graphs, communication complexity, ACC-circuits}
}
Document
DOI: 10.4230/DagSemProc.06111.22
Very Large Cliques are Easy to Detect

Authors: Alexander E. Andreev and Stasys Jukna

Published in: Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

Abstract
It is known that, for every constant $kgeq 3$, the presence of a $k$-clique (a complete subgraph on $k$ vertices) in an $n$-vertex graph cannot be detected by a monotone boolean circuit using fewer than $Omega((n/log n)^k)$ gates. We show that, for every constant $k$, the presence of an $(n-k)$-clique in an $n$-vertex graph can be detected by a monotone circuit using only $O(n^2log n)$ gates. Moreover, if we allow unbounded fanin, then $O(log n)$ gates are enough.
Cite as

Alexander E. Andreev and Stasys Jukna. Very Large Cliques are Easy to Detect. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{andreev_et_al:DagSemProc.06111.22,
  author =	{Andreev, Alexander E. and Jukna, Stasys},
  title =	{{Very Large Cliques are Easy to Detect}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--7},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.22},
  URN =		{urn:nbn:de:0030-drops-6092},
  doi =		{10.4230/DagSemProc.06111.22},
  annote =	{Keywords: Clique function, monotone circuits, perfect hashing}
}
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