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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

A circuit 𝒞 samples a distribution X with an error ε if the statistical distance between the output of 𝒞 on the uniform input and X is ε. We study the hardness of sampling a uniform distribution over the set of n-bit strings of Hamming weight k denoted by Uⁿ_k for decision forests, i.e. every output bit is computed as a decision tree of the inputs. For every k there is an O(log n)-depth decision forest sampling Uⁿ_k with an inverse-polynomial error [Emanuele Viola, 2012; Czumaj, 2015]. We show that for every ε > 0 there exists τ such that for decision depth τ log (n/k) / log log (n/k), the error for sampling U_kⁿ is at least 1-ε. Our result is based on the recent robust sunflower lemma [Ryan Alweiss et al., 2021; Rao, 2019].
Our second result is about matching a set of n-bit strings with the image of a d-local circuit, i.e. such that each output bit depends on at most d input bits. We study the set of all n-bit strings whose Hamming weight is at least n/2. We improve the previously known locality lower bound from Ω(log^* n) [Beyersdorff et al., 2013] to Ω(√log n), leaving only a quartic gap from the best upper bound of O(log² n).

Yuval Filmus, Itai Leigh, Artur Riazanov, and Dmitry Sokolov. Sampling and Certifying Symmetric Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{filmus_et_al:LIPIcs.APPROX/RANDOM.2023.36, author = {Filmus, Yuval and Leigh, Itai and Riazanov, Artur and Sokolov, Dmitry}, title = {{Sampling and Certifying Symmetric Functions}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {36:1--36:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.36}, URN = {urn:nbn:de:0030-drops-188611}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.36}, annote = {Keywords: sampling, lower bounds, robust sunflowers, decision trees, switching networks} }

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**Published in:** LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Following the paper of Alekhnovich, Ben-Sasson, Razborov, Wigderson [Michael Alekhnovich et al., 2004] we call a pseudorandom generator ℱ:{0, 1}ⁿ → {0, 1}^m hard for a propositional proof system P if P cannot efficiently prove the (properly encoded) statement b ∉ Im(ℱ) for any string b ∈ {0, 1}^m.
In [Michael Alekhnovich et al., 2004] the authors suggested the "functional encoding" of the considered statement for Nisan-Wigderson generator that allows the introduction of "local" extension variables. These extension variables may potentially significantly increase the power of the proof system. In [Michael Alekhnovich et al., 2004] authors gave a lower bound of exp[Ω(n²/{m⋅2^{2^Δ}})] on the length of Resolution proofs where Δ is the degree of the dependency graph of the generator. This lower bound meets the barrier for the restriction technique.
In this paper, we introduce a "heavy width" measure for Resolution that allows us to show a lower bound of exp[n²/{m 2^𝒪(εΔ)}] on the length of Resolution proofs of the considered statement for the Nisan-Wigderson generator. This gives an exponential lower bound up to Δ := log^{2 - δ} n (the bigger degree the more extension variables we can use). In [Michael Alekhnovich et al., 2004] authors left an open problem to get rid of scaling factor 2^{2^Δ}, it is a solution to this open problem.

Dmitry Sokolov. Pseudorandom Generators, Resolution and Heavy Width. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{sokolov:LIPIcs.CCC.2022.15, author = {Sokolov, Dmitry}, title = {{Pseudorandom Generators, Resolution and Heavy Width}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {15:1--15:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.15}, URN = {urn:nbn:de:0030-drops-165770}, doi = {10.4230/LIPIcs.CCC.2022.15}, annote = {Keywords: proof complexity, pseudorandom generators, resolution, lower bounds} }

Document

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

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].

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

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

Semialgebraic proof systems have been studied extensively in proof complexity since the late 1990s to understand the power of Gröbner basis computations, linear and semidefinite programming hierarchies, and other methods. Such proof systems are defined alternately with only the original variables of the problem and with special formal variables for positive and negative literals, but there seems to have been no study how these different definitions affect the power of the proof systems. We show for Nullstellensatz, polynomial calculus, Sherali-Adams, and sums-of-squares that adding formal variables for negative literals makes the proof systems exponentially stronger, with respect to the number of terms in the proofs. These separations are witnessed by CNF formulas that are easy for resolution, which establishes that polynomial calculus, Sherali-Adams, and sums-of-squares cannot efficiently simulate resolution without having access to variables for negative literals.

Susanna F. de Rezende, Massimo Lauria, Jakob Nordström, and Dmitry Sokolov. The Power of Negative Reasoning. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 40:1-40:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{derezende_et_al:LIPIcs.CCC.2021.40, author = {de Rezende, Susanna F. and Lauria, Massimo and Nordstr\"{o}m, Jakob and Sokolov, Dmitry}, title = {{The Power of Negative Reasoning}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {40:1--40:24}, 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.2021.40}, URN = {urn:nbn:de:0030-drops-143140}, doi = {10.4230/LIPIcs.CCC.2021.40}, annote = {Keywords: Proof complexity, Polynomial calculus, Nullstellensatz, Sums-of-squares, Sherali-Adams} }

Document

**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense graphs as in [Raz '04] and [Razborov '03, '04]. We obtain our results by revisiting Razborov’s pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.

Susanna F. de Rezende, Jakob Nordström, Kilian Risse, and Dmitry Sokolov. Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 28:1-28:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{derezende_et_al:LIPIcs.CCC.2020.28, author = {de Rezende, Susanna F. and Nordstr\"{o}m, Jakob and Risse, Kilian and Sokolov, Dmitry}, title = {{Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {28:1--28:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.28}, URN = {urn:nbn:de:0030-drops-125804}, doi = {10.4230/LIPIcs.CCC.2020.28}, annote = {Keywords: proof complexity, resolution, weak pigeonhole principle, perfect matching, sparse graphs} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Building on [Clegg et al. '96], [Impagliazzo et al. '99] established that if an unsatisfiable k-CNF formula over n variables has a refutation of size S in the polynomial calculus resolution proof system, then this formula also has a refutation of degree k + O(√(n log S)). The proof of this works by converting a small-size refutation into a small-degree one, but at the expense of increasing the proof size exponentially. This raises the question of whether it is possible to achieve both small size and small degree in the same refutation, or whether the exponential blow-up is inherent. Using and extending ideas from [Thapen '16], who studied the analogous question for the resolution proof system, we prove that a strong size-degree trade-off is necessary.

Guillaume Lagarde, Jakob Nordström, Dmitry Sokolov, and Joseph Swernofsky. Trade-Offs Between Size and Degree in Polynomial Calculus. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 72:1-72:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lagarde_et_al:LIPIcs.ITCS.2020.72, author = {Lagarde, Guillaume and Nordstr\"{o}m, Jakob and Sokolov, Dmitry and Swernofsky, Joseph}, title = {{Trade-Offs Between Size and Degree in Polynomial Calculus}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {72:1--72:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.72}, URN = {urn:nbn:de:0030-drops-117573}, doi = {10.4230/LIPIcs.ITCS.2020.72}, annote = {Keywords: proof complexity, polynomial calculus, polynomial calculus resolution, PCR, size-degree trade-off, resolution, colored polynomial local search} }

Document

**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Separations: We introduce a monotone variant of Xor-Sat and show it has exponential monotone circuit complexity. Since Xor-Sat is in NC^2, this improves qualitatively on the monotone vs. non-monotone separation of Tardos (1988). We also show that monotone span programs over R can be exponentially more powerful than over finite fields. These results can be interpreted as separating subclasses of TFNP in communication complexity.
Characterizations: We show that the communication (resp. query) analogue of PPA (subclass of TFNP) captures span programs over F_2 (resp. Nullstellensatz degree over F_2). Previously, it was known that communication FP captures formulas (Karchmer - Wigderson, 1988) and that communication PLS captures circuits (Razborov, 1995).

Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in Monotone Complexity and TFNP. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 38:1-38:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{goos_et_al:LIPIcs.ITCS.2019.38, author = {G\"{o}\"{o}s, Mika and Kamath, Pritish and Robere, Robert and Sokolov, Dmitry}, title = {{Adventures in Monotone Complexity and TFNP}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {38:1--38:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.38}, URN = {urn:nbn:de:0030-drops-101316}, doi = {10.4230/LIPIcs.ITCS.2019.38}, annote = {Keywords: TFNP, Monotone Complexity, Communication Complexity, Proof Complexity} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

Atserias, Kolaitis, and Vardi showed that the proof system of Ordered Binary Decision Diagrams with conjunction and weakening, OBDD(^, weakening), simulates CP^* (Cutting Planes with unary coefficients). We show that OBDD(^, weakening) can give exponentially shorter proofs than dag-like cutting planes. This is proved by showing that the Clique-Coloring tautologies have polynomial size proofs in the OBDD(^, weakening) system.
The reordering rule allows changing the variable order for OBDDs. We show that OBDD(^, weakening, reordering) is strictly stronger than OBDD(^, weakening). This is proved using the Clique-Coloring tautologies, and by transforming tautologies using coded permutations and orification. We also give CNF formulas which have polynomial size OBDD(^) proofs but require superpolynomial (actually, quasipolynomial size) resolution proofs, and thus we partially resolve an open question proposed by Groote and Zantema.
Applying dag-like and tree-like lifting techniques to the mentioned results, we completely analyze which of the systems among CP^*, OBDD(^), OBDD(^, reordering), OBDD(^, weakening) and OBDD(^, weakening, reordering) polynomially simulate each other. For dag-like proof systems, some of our separations are quasipolynomial and some are exponential; for tree-like systems, all of our separations are exponential.

Sam Buss, Dmitry Itsykson, Alexander Knop, and Dmitry Sokolov. Reordering Rule Makes OBDD Proof Systems Stronger. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 16:1-16:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{buss_et_al:LIPIcs.CCC.2018.16, author = {Buss, Sam and Itsykson, Dmitry and Knop, Alexander and Sokolov, Dmitry}, title = {{Reordering Rule Makes OBDD Proof Systems Stronger}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {16:1--16:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.16}, URN = {urn:nbn:de:0030-drops-88720}, doi = {10.4230/LIPIcs.CCC.2018.16}, annote = {Keywords: Proof complexity, OBDD, Tseitin formulas, the Clique-Coloring principle, lifting theorems} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

In 2004 Atserias, Kolaitis and Vardi proposed OBDD-based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of identically false OBDD from OBDDs representing clauses of the initial formula. All OBDDs in such proofs have the same order of variables. We initiate the study of OBDD based proof systems that additionally contain a rule that allows to change the order in OBDDs. At first we consider a proof system OBDD(and, reordering) that uses the conjunction (join) rule and the rule that allows to change the order. We exponentially separate this proof system from OBDD(and)-proof system that uses only the conjunction rule. We prove two exponential lower bounds on the size of OBDD(and, reordering)-refutations of Tseitin formulas and the pigeonhole principle. The first lower bound was previously unknown even for OBDD(and)-proofs and the second one extends the result of Tveretina et al. from OBDD(and) to OBDD(and, reordering).
In 2004 Pan and Vardi proposed an approach to the propositional satisfiability problem based on OBDDs and symbolic quantifier elimination (we denote algorithms based on this approach as OBDD(and, exists)-algorithms. We notice that there exists an OBDD(and, exists)-algorithm that solves satisfiable and unsatisfiable Tseitin formulas in polynomial time. In contrast, we show that there exist formulas representing systems of linear equations over F_2 that are hard for OBDD(and, exists, reordering)-algorithms. Our hard instances are satisfiable formulas representing systems of linear equations over F_2 that correspond to some checksum matrices of error correcting codes.

Dmitry Itsykson, Alexander Knop, Andrey Romashchenko, and Dmitry Sokolov. On OBDD-Based Algorithms and Proof Systems That Dynamically Change Order of Variables. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 43:1-43:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{itsykson_et_al:LIPIcs.STACS.2017.43, author = {Itsykson, Dmitry and Knop, Alexander and Romashchenko, Andrey and Sokolov, Dmitry}, title = {{On OBDD-Based Algorithms and Proof Systems That Dynamically Change Order of Variables}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {43:1--43:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.43}, URN = {urn:nbn:de:0030-drops-69914}, doi = {10.4230/LIPIcs.STACS.2017.43}, annote = {Keywords: Proof complexity, OBDD, error-correcting codes, Tseitin formulas, expanders} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

We address the following question in the average-case complexity: does there exists a language L such that for all easy distributions D the distributional problem (L, D) is easy on the average while there exists some more hard distribution D' such that (L, D') is hard on the average? We consider two complexity measures of distributions: the complexity of sampling and the complexity of computing the distribution function.
For the complexity of sampling of distribution, we establish a connection between the above question and the hierarchy theorem for sampling distribution recently studied by Thomas Watson. Using this connection we prove that for every 0 < a < b there exist a language L, an ensemble of distributions D samplable in n^{log^b n} steps and a linear-time algorithm A such that for every ensemble of distribution F that samplable in n^{log^a n} steps, A correctly decides L on all inputs from {0, 1}^n except for a set that has infinitely small F-measure, and for every algorithm B there are infinitely many n such that the set of all elements of {0, 1}^n for which B correctly decides L has infinitely small D-measure.
In case of complexity of computing the distribution function we prove the following tight result: for every a > 0 there exist a language L, an ensemble of polynomial-time computable distributions D, and a linear-time algorithm A such that for every computable in n^a steps ensemble of distributions F , A correctly decides L on all inputs from {0, 1}^n except for a set that has F-measure at most 2^{-n/2} , and for every algorithm B there are infinitely many n such that the set of all elements of {0, 1}^n for which B correctly decides L has D-measure at most 2^{-n+1}.

Dmitry Itsykson, Alexander Knop, and Dmitry Sokolov. Complexity of Distributions and Average-Case Hardness. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 38:1-38:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{itsykson_et_al:LIPIcs.ISAAC.2016.38, author = {Itsykson, Dmitry and Knop, Alexander and Sokolov, Dmitry}, title = {{Complexity of Distributions and Average-Case Hardness}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {38:1--38:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.38}, URN = {urn:nbn:de:0030-drops-68083}, doi = {10.4230/LIPIcs.ISAAC.2016.38}, annote = {Keywords: average-case complexity, hierarchy theorem, sampling distributions, diagonalization} }

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