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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

The minimum number of clauses in a CNF representation of the parity function x₁ ⊕ x₂ ⊕ … ⊕ x_n is 2^{n-1}. One can obtain a more compact CNF encoding by using non-deterministic variables (also known as guess or auxiliary variables). In this paper, we prove the following lower bounds, that almost match known upper bounds, on the number m of clauses and the maximum width k of clauses: 1) if there are at most s auxiliary variables, then m ≥ Ω(2^{n/(s+1)}/n) and k ≥ n/(s+1); 2) the minimum number of clauses is at least 3n. We derive the first two bounds from the Satisfiability Coding Lemma due to Paturi, Pudlák, and Zane using a tight connection between CNF encodings and depth-3 circuits. In particular, we show that lower bounds on the size of a CNF encoding of a Boolean function imply depth-3 circuit lower bounds for this function.

Gregory Emdin, Alexander S. Kulikov, Ivan Mihajlin, and Nikita Slezkin. CNF Encodings of Parity. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 47:1-47:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{emdin_et_al:LIPIcs.MFCS.2022.47, author = {Emdin, Gregory and Kulikov, Alexander S. and Mihajlin, Ivan and Slezkin, Nikita}, title = {{CNF Encodings of Parity}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {47:1--47:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.47}, URN = {urn:nbn:de:0030-drops-168455}, doi = {10.4230/LIPIcs.MFCS.2022.47}, annote = {Keywords: encoding, parity, lower bounds, circuits, CNF} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Finding exact circuit size is notoriously hard. Whereas modern computers and algorithmic techniques allow to find a circuit of size seven in the blink of an eye, it may take more than a week to search for a circuit of size thirteen. One of the reasons of this behavior is that the search space is enormous: the number of circuits of size s is s^Θ(s), the number of Boolean functions on n variables is 2^(2ⁿ).
In this paper, we explore the following natural heuristic idea for decreasing the size of a given circuit: go through all its subcircuits of moderate size and check whether any of them can be improved by reducing to SAT. This may be viewed as a local search approach: we search for a smaller circuit in a ball around a given circuit. Through this approach, we prove new upper bounds on the circuit size of various symmetric functions. We also demonstrate that some upper bounds that were proved by hand decades ago, can nowadays be found automatically in a few seconds.

Alexander S. Kulikov, Danila Pechenev, and Nikita Slezkin. SAT-Based Circuit Local Improvement. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 67:1-67:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kulikov_et_al:LIPIcs.MFCS.2022.67, author = {Kulikov, Alexander S. and Pechenev, Danila and Slezkin, Nikita}, title = {{SAT-Based Circuit Local Improvement}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {67:1--67:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.67}, URN = {urn:nbn:de:0030-drops-168659}, doi = {10.4230/LIPIcs.MFCS.2022.67}, annote = {Keywords: circuits, algorithms, complexity theory, SAT, SAT solvers, heuristics} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

In the minimum common string partition problem (MCSP), one gets two strings and is asked to find the minimum number of cuts in the first string such that the second string can be obtained by rearranging the resulting pieces. It is a difficult algorithmic problem having applications in computational biology, text processing, and data compression. MCSP has been studied extensively from various algorithmic angles: there are many papers studying approximation, heuristic, and parameterized algorithms. At the same time, almost nothing is known about its exact complexity. In this paper, we present new results in this direction. We improve the known 2ⁿ upper bound (where n is the length of input strings) to ϕⁿ where ϕ ≈ 1.618... is the golden ratio. The algorithm uses Fibonacci numbers to encode subsets as monomials of a certain implicit polynomial and extracts one of its coefficients using the fast Fourier transform. Then, we show that the case of constant size alphabet can be solved in subexponential time 2^{O(nlog log n/log n)} by a hybrid strategy: enumerate all long pieces and use dynamic programming over histograms of short pieces. Finally, we prove almost matching lower bounds assuming the Exponential Time Hypothesis.

Marek Cygan, Alexander S. Kulikov, Ivan Mihajlin, Maksim Nikolaev, and Grigory Reznikov. Minimum Common String Partition: Exact Algorithms. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cygan_et_al:LIPIcs.ESA.2021.35, author = {Cygan, Marek and Kulikov, Alexander S. and Mihajlin, Ivan and Nikolaev, Maksim and Reznikov, Grigory}, title = {{Minimum Common String Partition: Exact Algorithms}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {35:1--35:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.35}, URN = {urn:nbn:de:0030-drops-146167}, doi = {10.4230/LIPIcs.ESA.2021.35}, annote = {Keywords: similarity measure, string distance, exact algorithms, upper bounds, lower bounds} }

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

The best known size lower bounds against unrestricted circuits have remained around 3n for several decades. Moreover, the only known technique for proving lower bounds in this model, gate elimination, is inherently limited to proving lower bounds of less than 5n. In this work, we propose a non-gate-elimination approach for obtaining circuit lower bounds, via certain depth-three lower bounds. We prove that every (unbounded-depth) circuit of size s can be expressed as an OR of 2^{s/3.9} 16-CNFs. For DeMorgan formulas, the best known size lower bounds have been stuck at around n^{3-o(1)} for decades. Under a plausible hypothesis about probabilistic polynomials, we show that n^{4-ε}-size DeMorgan formulas have 2^{n^{1-Ω(ε)}}-size depth-3 circuits which are approximate sums of n^{1-Ω(ε)}-degree polynomials over F₂. While these structural results do not immediately lead to new lower bounds, they do suggest new avenues of attack on these longstanding lower bound problems.
Our results complement the classical depth-3 reduction results of Valiant, which show that logarithmic-depth circuits of linear size can be computed by an OR of 2^{ε n} n^δ-CNFs, and slightly stronger results for series-parallel circuits. It is known that no purely graph-theoretic reduction could yield interesting depth-3 circuits from circuits of super-logarithmic depth. We overcome this limitation (for small-size circuits) by taking into account both the graph-theoretic and functional properties of circuits and formulas.
We show that improvements of the following pseudorandom constructions imply super-linear circuit lower bounds for log-depth circuits via Valiant’s reduction: dispersers for varieties, correlation with constant degree polynomials, matrix rigidity, and hardness for depth-3 circuits with constant bottom fan-in. On the other hand, our depth reductions show that even modest improvements of the known constructions give elementary proofs of improved (but still linear) circuit lower bounds.

Alexander Golovnev, Alexander S. Kulikov, and R. Ryan Williams. Circuit Depth Reductions. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{golovnev_et_al:LIPIcs.ITCS.2021.24, author = {Golovnev, Alexander and Kulikov, Alexander S. and Williams, R. Ryan}, title = {{Circuit Depth Reductions}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {24:1--24:20}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.24}, URN = {urn:nbn:de:0030-drops-135633}, doi = {10.4230/LIPIcs.ITCS.2021.24}, annote = {Keywords: Circuit complexity, formula complexity, pseudorandomness, matrix rigidity} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Let A in {0,1}^{n x n} be a matrix with z zeroes and u ones and x be an n-dimensional vector of formal variables over a semigroup (S, o). How many semigroup operations are required to compute the linear operator Ax?
As we observe in this paper, this problem contains as a special case the well-known range queries problem and has a rich variety of applications in such areas as graph algorithms, functional programming, circuit complexity, and others. It is easy to compute Ax using O(u) semigroup operations. The main question studied in this paper is: can Ax be computed using O(z) semigroup operations? We prove that in general this is not possible: there exists a matrix A in {0,1}^{n x n} with exactly two zeroes in every row (hence z=2n) whose complexity is Theta(n alpha(n)) where alpha(n) is the inverse Ackermann function. However, for the case when the semigroup is commutative, we give a constructive proof of an O(z) upper bound. This implies that in commutative settings, complements of sparse matrices can be processed as efficiently as sparse matrices (though the corresponding algorithms are more involved). Note that this covers the cases of Boolean and tropical semirings that have numerous applications, e.g., in graph theory.
As a simple application of the presented linear-size construction, we show how to multiply two n x n matrices over an arbitrary semiring in O(n^2) time if one of these matrices is a 0/1-matrix with O(n) zeroes (i.e., a complement of a sparse matrix).

Alexander S. Kulikov, Ivan Mikhailin, Andrey Mokhov, and Vladimir Podolskii. Complexity of Linear Operators. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 17:1-17:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kulikov_et_al:LIPIcs.ISAAC.2019.17, author = {Kulikov, Alexander S. and Mikhailin, Ivan and Mokhov, Andrey and Podolskii, Vladimir}, title = {{Complexity of Linear Operators}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {17:1--17:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.17}, URN = {urn:nbn:de:0030-drops-115137}, doi = {10.4230/LIPIcs.ISAAC.2019.17}, annote = {Keywords: algorithms, linear operators, commutativity, range queries, circuit complexity, lower bounds, upper bounds} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits 2 11/23-approximation in polynomial time (Mucha, SODA'13). While this algorithm and its analysis are technically involved, the 30 years old Greedy Conjecture claims that the trivial and efficient Greedy Algorithm gives a 2-approximation for SCS.
We develop a graph-theoretic framework for studying approximation algorithms for SCS. The framework is reminiscent of the classical 2-approximation for Traveling Salesman: take two copies of an optimal solution, apply a trivial edge-collapsing procedure, and get an approximate solution. In this framework, we observe two surprising properties of SCS solutions, and we conjecture that they hold for all input instances. The first conjecture, that we call Collapsing Superstring conjecture, claims that there is an elementary way to transform any solution repeated twice into the same graph G. This conjecture would give an elementary 2-approximate algorithm for SCS. The second conjecture claims that not only the resulting graph G is the same for all solutions, but that G can be computed by an elementary greedy procedure called Greedy Hierarchical Algorithm.
While the second conjecture clearly implies the first one, perhaps surprisingly we prove their equivalence. We support these equivalent conjectures by giving a proof for the special case where all input strings have length at most 3 (which until recently had been the only case where the Greedy Conjecture was proven). We also tested our conjectures on millions of instances of SCS.
We prove that the standard Greedy Conjecture implies Greedy Hierarchical Conjecture, while the latter is sufficient for an efficient greedy 2-approximate approximation of SCS. Except for its (conjectured) good approximation ratio, the Greedy Hierarchical Algorithm provably finds a 3.5-approximation, and finds exact solutions for the special cases where we know polynomial time (not greedy) exact algorithms: (1) when the input strings form a spectrum of a string (2) when all input strings have length at most 2.

Alexander Golovnev, Alexander S. Kulikov, Alexander Logunov, Ivan Mihajlin, and Maksim Nikolaev. Collapsing Superstring Conjecture. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 26:1-26:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{golovnev_et_al:LIPIcs.APPROX-RANDOM.2019.26, author = {Golovnev, Alexander and Kulikov, Alexander S. and Logunov, Alexander and Mihajlin, Ivan and Nikolaev, Maksim}, title = {{Collapsing Superstring Conjecture}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {26:1--26:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.26}, URN = {urn:nbn:de:0030-drops-112411}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.26}, annote = {Keywords: superstring, shortest common superstring, approximation, greedy algorithms, greedy conjecture} }

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

We study the following computational problem: for which values of k, the majority of n bits MAJ_n can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJ_k o MAJ_k. We observe that the minimum value of k for which there exists a MAJ_k o MAJ_k circuit that has high correlation with the majority of n bits is equal to Theta(sqrt(n)). We then show that for a randomized MAJ_k o MAJ_k circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n^(2/3+o(1)). We show a worst case lower bound: if a MAJ_k o MAJ_k circuit computes the majority of n bits correctly on all inputs, then k <= n^(13/19+o(1)). This lower bound exceeds the optimal value for randomized circuits and thus is unreachable for pure randomized techniques. For depth 3 circuits we show that a circuit with k= O(n^(2/3)) can compute MAJ_n correctly on all inputs.

Alexander S. Kulikov and Vladimir V. Podolskii. Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kulikov_et_al:LIPIcs.STACS.2017.49, author = {Kulikov, Alexander S. and Podolskii, Vladimir V.}, title = {{Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {49:1--49:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.49}, URN = {urn:nbn:de:0030-drops-69832}, doi = {10.4230/LIPIcs.STACS.2017.49}, annote = {Keywords: circuit complexity, computational complexity, threshold, majority, lower bound, upper bound} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function.
In this paper, we provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT that is built on ideas of Chen and Kabanets. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount. This case analysis immediately implies an upper bound for #SAT. To~obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration.
We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework.
Using this framework, we prove the following new bounds: average case lower bounds of 3.24n and 2.59n for circuits over U_2 and B_2, respectively (though the lower bound for the basis B_2 is given for a quadratic disperser whose explicit construction is not currently known), and faster than 2^n #SAT-algorithms for circuits over U_2 and B_2 of size at most 3.24n and 2.99n, respectively. Here by B_2 we mean the set of all bivariate Boolean functions, and by U_2 the set of all bivariate Boolean functions except for parity and its complement.

Alexander Golovnev, Alexander S. Kulikov, Alexander V. Smal, and Suguru Tamaki. Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{golovnev_et_al:LIPIcs.MFCS.2016.45, author = {Golovnev, Alexander and Kulikov, Alexander S. and Smal, Alexander V. and Tamaki, Suguru}, title = {{Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {45:1--45:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.45}, URN = {urn:nbn:de:0030-drops-64588}, doi = {10.4230/LIPIcs.MFCS.2016.45}, annote = {Keywords: circuit complexity, lower bounds, exponential time algorithms, satisfiability} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Although a simple counting argument shows the existence of Boolean functions of exponential circuit complexity, proving superlinear circuit lower bounds for explicit functions seems to be out of reach of the current techniques. There has been a (very slow) progress in proving linear lower bounds with the latest record of 3 1/86*n-o(n). All known lower bounds are based on the so-called gate elimination technique. A typical gate elimination argument shows that it is possible to eliminate several gates from an optimal circuit by making one or several substitutions to the input variables and repeats this inductively. In this note we prove that this method cannot achieve linear bounds of cn beyond a certain constant c, where c depends only on the number of substitutions made at a single step of the induction.

Alexander Golovnev, Edward A. Hirsch, Alexander Knop, and Alexander S. Kulikov. On the Limits of Gate Elimination. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{golovnev_et_al:LIPIcs.MFCS.2016.46, author = {Golovnev, Alexander and Hirsch, Edward A. and Knop, Alexander and Kulikov, Alexander S.}, title = {{On the Limits of Gate Elimination}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {46:1--46:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.46}, URN = {urn:nbn:de:0030-drops-64593}, doi = {10.4230/LIPIcs.MFCS.2016.46}, annote = {Keywords: circuit complexity, lower bounds, gate elimination} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

The Secluded Path problem introduced by Chechik et al. in [ESA 2013] models a situation where a sensitive information has to be transmitted between a pair of nodes along a path in a network.
The measure of the quality of a selected path is its exposure, which is the total weight of vertices in its closed neighborhood. In order to minimize the risk of intercepting the information, we are interested in selecting a secluded path, i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure of the tree is minimized. In this work, we obtain the following results about parameterized complexity of secluded connectivity problems.
We start from an observation that being parameterized by the size of the exposure, the problem is fixed-parameter tractable (FPT). More precisely, we give an algorithm deciding if a graph G with a given cost function w:V(G)->N contains a secluded path of exposure at most k with the cost at most C in time O(3^{k/3}(n+m) log W), where W is the maximum value of w on an input graph G. Similarly,
Secluded Steiner Tree is solvable in time O(2^{k}k^2 (n+m) log W).
The main result of this paper is about "above guarantee" parameterizations for secluded problems. We show that Secluded Steiner Tree is FPT being parameterized by r+p, where p is the number of the terminals, l the size of an optimum Steiner tree, and r=k-l. We complement this result by showing that the problem is co-W[1]-hard when parameterized by r only.
We also investigate Secluded Steiner Tree from kernelization perspective and provide several lower and upper bounds when parameters are the treewidth, the size of a vertex cover, maximum vertex degree and the solution size. Finally, we refine the algorithmic result of Chechik et al. by improving the exponential dependence from the treewidth of the input graph.

Fedor V. Fomin, Petr A. Golovach, Nikolay Karpov, and Alexander S. Kulikov. Parameterized Complexity of Secluded Connectivity Problems. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 408-419, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{fomin_et_al:LIPIcs.FSTTCS.2015.408, author = {Fomin, Fedor V. and Golovach, Petr A. and Karpov, Nikolay and Kulikov, Alexander S.}, title = {{Parameterized Complexity of Secluded Connectivity Problems}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {408--419}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.408}, URN = {urn:nbn:de:0030-drops-56318}, doi = {10.4230/LIPIcs.FSTTCS.2015.408}, annote = {Keywords: Secluded path, Secluded Steiner tree, parameterized complexity} }

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