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**Published in:** LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

For every n, we construct a sum-of-squares identity (∑_{i=1}^n x_i²) (∑_{j=1}^n y_j²) = ∑_{k=1}^s f_k², where f_k are bilinear forms with complex coefficients and s = O(n^1.62). Previously, such a construction was known with s = O(n²/log n). The same bound holds over any field of positive characteristic.

Pavel Hrubeš. A Subquadratic Upper Bound on Sum-Of-Squares Composition Formulas. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 12:1-12:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hrubes:LIPIcs.CCC.2024.12, author = {Hrube\v{s}, Pavel}, title = {{A Subquadratic Upper Bound on Sum-Of-Squares Composition Formulas}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {12:1--12:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-331-7}, ISSN = {1868-8969}, year = {2024}, volume = {300}, editor = {Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.12}, URN = {urn:nbn:de:0030-drops-204082}, doi = {10.4230/LIPIcs.CCC.2024.12}, annote = {Keywords: Sum-of-squares composition formulas, Hurwitz’s problem, non-commutative arithmetic circuit} }

Document

**Published in:** LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Given a non-negative real matrix M of non-negative rank at least r, can we witness this fact by a small submatrix of M? While Moitra (SIAM J. Comput. 2013) proved that this cannot be achieved exactly, we show that such a witnessing is possible approximately: an m×n matrix of non-negative rank r always contains a submatrix with at most r³ rows and columns with non-negative rank at least Ω(r/(log n log m)). A similar result is proved for the 1-partition number of a Boolean matrix and, consequently, also for its two-player deterministic communication complexity. Tightness of the latter estimate is closely related to the log-rank conjecture of Lovász and Saks.

Pavel Hrubeš. Hard Submatrices for Non-Negative Rank and Communication Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 13:1-13:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hrubes:LIPIcs.CCC.2024.13, author = {Hrube\v{s}, Pavel}, title = {{Hard Submatrices for Non-Negative Rank and Communication Complexity}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {13:1--13:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-331-7}, ISSN = {1868-8969}, year = {2024}, volume = {300}, editor = {Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.13}, URN = {urn:nbn:de:0030-drops-204097}, doi = {10.4230/LIPIcs.CCC.2024.13}, annote = {Keywords: Non-negative rank, communication complexity, extension complexity} }

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**Published in:** LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree d which requires homogeneous non-commutative circuit of size Ω(d/log d). For an n-variate polynomial with n > 1, the result can be improved to Ω(nd), if d ≤ n, or Ω(nd (log n)/(log d)), if d ≥ n. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.

Prerona Chatterjee and Pavel Hrubeš. New Lower Bounds Against Homogeneous Non-Commutative Circuits. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 13:1-13:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2023.13, author = {Chatterjee, Prerona and Hrube\v{s}, Pavel}, title = {{New Lower Bounds Against Homogeneous Non-Commutative Circuits}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {13:1--13:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.13}, URN = {urn:nbn:de:0030-drops-182835}, doi = {10.4230/LIPIcs.CCC.2023.13}, annote = {Keywords: Algebraic circuit complexity, Non-Commutative Circuits, Homogeneous Computation, Lower bounds against algebraic circuits} }

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

We define the shadow complexity of a polytope P as the maximum number of vertices in a linear projection of P to the plane. We describe connections to algebraic complexity and to parametrized optimization. We also provide several basic examples and constructions, and develop tools for bounding shadow complexity.

Pavel Hrubeš and Amir Yehudayoff. Shadows of Newton Polytopes. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 9:1-9:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hrubes_et_al:LIPIcs.CCC.2021.9, author = {Hrube\v{s}, Pavel and Yehudayoff, Amir}, title = {{Shadows of Newton Polytopes}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {9:1--9:23}, 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.2021.9}, URN = {urn:nbn:de:0030-drops-142833}, doi = {10.4230/LIPIcs.CCC.2021.9}, annote = {Keywords: Newton polytope, Monotone arithmetic circuit} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets S_1,...,S_k subset [n] is balancing if for every subset X subset {1,2,...,n} of size n/2, there is an i in [k] so that |S_i cap X| = |S_i|/2. We extend and simplify the framework developed by Hegedűs for proving lower bounds on the size of balancing set families. We prove that if n=2p for a prime p, then k >= p. For arbitrary values of n, we show that k >= n/2 - o(n).
We then exploit the connection between balancing families and depth-2 threshold circuits. This connection helps resolve a question raised by Kulikov and Podolskii on the fan-in of depth-2 majority circuits computing the majority function on n bits. We show that any depth-2 threshold circuit that computes the majority on n bits has at least one gate with fan-in at least n/2 - o(n). We also prove a sharp lower bound on the fan-in of depth-2 threshold circuits computing a specific weighted threshold function.

Pavel Hrubeš, Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao, and Amir Yehudayoff. Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 72:1-72:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hrubes_et_al:LIPIcs.ICALP.2019.72, author = {Hrube\v{s}, Pavel and Natarajan Ramamoorthy, Sivaramakrishnan and Rao, Anup and Yehudayoff, Amir}, title = {{Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {72:1--72:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.72}, URN = {urn:nbn:de:0030-drops-106487}, doi = {10.4230/LIPIcs.ICALP.2019.72}, annote = {Keywords: Balancing sets, depth-2 threshold circuits, polynomials, majority, weighted thresholds} }

Document

**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

The isoperimetric profile of a graph is a function that measures, for an integer k, the size of the smallest edge boundary over all sets of vertices of size k. We observe a connection between isoperimetric profiles and computational complexity. We illustrate this connection by an example from communication complexity, but our main result is in algebraic complexity.
We prove a sharp super-polynomial separation between monotone arithmetic circuits and monotone arithmetic branching programs. This shows that the classical simulation of arithmetic circuits by arithmetic branching programs by Valiant, Skyum, Berkowitz, and Rackoff (1983) cannot be improved, as long as it preserves monotonicity.
A key ingredient in the proof is an accurate analysis of the isoperimetric profile of finite full binary trees. We show that the isoperimetric profile of a full binary tree constantly fluctuates between one and almost the depth of the tree.

Pavel Hrubes and Amir Yehudayoff. On Isoperimetric Profiles and Computational Complexity. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 89:1-89:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{hrubes_et_al:LIPIcs.ICALP.2016.89, author = {Hrubes, Pavel and Yehudayoff, Amir}, title = {{On Isoperimetric Profiles and Computational Complexity}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {89:1--89:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.89}, URN = {urn:nbn:de:0030-drops-61964}, doi = {10.4230/LIPIcs.ICALP.2016.89}, annote = {Keywords: Monotone computation, separations, communication complexity, isoperimetry} }

Document

**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

In this paper, we compare the strength of the semantic and syntactic version of the cutting planes proof system.
First, we show that the lower bound technique of Pudlák applies also to semantic cutting planes: the proof system has feasible interpolation via monotone real circuits, which gives an exponential lower bound on lengths of semantic cutting planes refutations.
Second, we show that semantic refutations are stronger than syntactic ones. In particular, we give a formula for which any refutation in syntactic cutting planes requires exponential length, while there is a polynomial length refutation in semantic cutting planes. In other words, syntactic cutting planes does not p-simulate semantic cutting planes. We also give two incompatible integer inequalities which require exponential length refutation in syntactic cutting planes.
Finally, we pose the following problem, which arises in connection with semantic inference of arity larger than two: can every multivariate non-decreasing real function be expressed as a composition of non-decreasing real functions in two variables?

Yuval Filmus, Pavel Hrubeš, and Massimo Lauria. Semantic Versus Syntactic Cutting Planes. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{filmus_et_al:LIPIcs.STACS.2016.35, author = {Filmus, Yuval and Hrube\v{s}, Pavel and Lauria, Massimo}, title = {{Semantic Versus Syntactic Cutting Planes}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {35:1--35:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.35}, URN = {urn:nbn:de:0030-drops-57367}, doi = {10.4230/LIPIcs.STACS.2016.35}, annote = {Keywords: proof complexity, cutting planes, lower bounds} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

We consider boolean circuits in which every gate may compute an arbitrary boolean function of k other gates, for a parameter k. We give an explicit function $f:{0,1}^n -> {0,1} that requires at least Omega(log^2(n)) non-input gates when k = 2n/3. When the circuit is restricted to being layered and depth 2, we prove a lower bound of n^(Omega(1)) on the number of non-input gates. When the circuit is a formula with gates of fan-in k, we give a lower bound Omega(n^2/k*log(n)) on the total number of gates.
Our model is connected to some well known approaches to proving lower bounds in complexity theory. Optimal lower bounds for the Number-On-Forehead model in communication complexity, or for bounded depth circuits in AC_0, or extractors for varieties over small fields would imply strong lower bounds in our model. On the other hand, new lower bounds for our model would prove new time-space tradeoffs for branching programs and impossibility results for (fan-in 2) circuits with linear size and logarithmic depth. In particular, our lower bound gives a different proof for a known time-space tradeoff for oblivious branching programs.

Pavel Hrubes and Anup Rao. Circuits with Medium Fan-In. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 381-391, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hrubes_et_al:LIPIcs.CCC.2015.381, author = {Hrubes, Pavel and Rao, Anup}, title = {{Circuits with Medium Fan-In}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {381--391}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.381}, URN = {urn:nbn:de:0030-drops-50528}, doi = {10.4230/LIPIcs.CCC.2015.381}, annote = {Keywords: Boolean circuit, Complexity, Communication Complexity} }

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**Published in:** LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

For every n, we construct a sum-of-squares identity (∑_{i=1}^n x_i²) (∑_{j=1}^n y_j²) = ∑_{k=1}^s f_k², where f_k are bilinear forms with complex coefficients and s = O(n^1.62). Previously, such a construction was known with s = O(n²/log n). The same bound holds over any field of positive characteristic.

Pavel Hrubeš. A Subquadratic Upper Bound on Sum-Of-Squares Composition Formulas. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 12:1-12:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hrubes:LIPIcs.CCC.2024.12, author = {Hrube\v{s}, Pavel}, title = {{A Subquadratic Upper Bound on Sum-Of-Squares Composition Formulas}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {12:1--12:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-331-7}, ISSN = {1868-8969}, year = {2024}, volume = {300}, editor = {Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.12}, URN = {urn:nbn:de:0030-drops-204082}, doi = {10.4230/LIPIcs.CCC.2024.12}, annote = {Keywords: Sum-of-squares composition formulas, Hurwitz’s problem, non-commutative arithmetic circuit} }

Document

**Published in:** LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Given a non-negative real matrix M of non-negative rank at least r, can we witness this fact by a small submatrix of M? While Moitra (SIAM J. Comput. 2013) proved that this cannot be achieved exactly, we show that such a witnessing is possible approximately: an m×n matrix of non-negative rank r always contains a submatrix with at most r³ rows and columns with non-negative rank at least Ω(r/(log n log m)). A similar result is proved for the 1-partition number of a Boolean matrix and, consequently, also for its two-player deterministic communication complexity. Tightness of the latter estimate is closely related to the log-rank conjecture of Lovász and Saks.

Pavel Hrubeš. Hard Submatrices for Non-Negative Rank and Communication Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 13:1-13:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hrubes:LIPIcs.CCC.2024.13, author = {Hrube\v{s}, Pavel}, title = {{Hard Submatrices for Non-Negative Rank and Communication Complexity}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {13:1--13:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-331-7}, ISSN = {1868-8969}, year = {2024}, volume = {300}, editor = {Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.13}, URN = {urn:nbn:de:0030-drops-204097}, doi = {10.4230/LIPIcs.CCC.2024.13}, annote = {Keywords: Non-negative rank, communication complexity, extension complexity} }

Document

**Published in:** LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree d which requires homogeneous non-commutative circuit of size Ω(d/log d). For an n-variate polynomial with n > 1, the result can be improved to Ω(nd), if d ≤ n, or Ω(nd (log n)/(log d)), if d ≥ n. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.

Prerona Chatterjee and Pavel Hrubeš. New Lower Bounds Against Homogeneous Non-Commutative Circuits. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 13:1-13:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2023.13, author = {Chatterjee, Prerona and Hrube\v{s}, Pavel}, title = {{New Lower Bounds Against Homogeneous Non-Commutative Circuits}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {13:1--13:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.13}, URN = {urn:nbn:de:0030-drops-182835}, doi = {10.4230/LIPIcs.CCC.2023.13}, annote = {Keywords: Algebraic circuit complexity, Non-Commutative Circuits, Homogeneous Computation, Lower bounds against algebraic circuits} }

Document

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

We define the shadow complexity of a polytope P as the maximum number of vertices in a linear projection of P to the plane. We describe connections to algebraic complexity and to parametrized optimization. We also provide several basic examples and constructions, and develop tools for bounding shadow complexity.

Pavel Hrubeš and Amir Yehudayoff. Shadows of Newton Polytopes. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 9:1-9:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hrubes_et_al:LIPIcs.CCC.2021.9, author = {Hrube\v{s}, Pavel and Yehudayoff, Amir}, title = {{Shadows of Newton Polytopes}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {9:1--9:23}, 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.2021.9}, URN = {urn:nbn:de:0030-drops-142833}, doi = {10.4230/LIPIcs.CCC.2021.9}, annote = {Keywords: Newton polytope, Monotone arithmetic circuit} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets S_1,...,S_k subset [n] is balancing if for every subset X subset {1,2,...,n} of size n/2, there is an i in [k] so that |S_i cap X| = |S_i|/2. We extend and simplify the framework developed by Hegedűs for proving lower bounds on the size of balancing set families. We prove that if n=2p for a prime p, then k >= p. For arbitrary values of n, we show that k >= n/2 - o(n).
We then exploit the connection between balancing families and depth-2 threshold circuits. This connection helps resolve a question raised by Kulikov and Podolskii on the fan-in of depth-2 majority circuits computing the majority function on n bits. We show that any depth-2 threshold circuit that computes the majority on n bits has at least one gate with fan-in at least n/2 - o(n). We also prove a sharp lower bound on the fan-in of depth-2 threshold circuits computing a specific weighted threshold function.

Pavel Hrubeš, Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao, and Amir Yehudayoff. Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 72:1-72:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hrubes_et_al:LIPIcs.ICALP.2019.72, author = {Hrube\v{s}, Pavel and Natarajan Ramamoorthy, Sivaramakrishnan and Rao, Anup and Yehudayoff, Amir}, title = {{Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {72:1--72:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.72}, URN = {urn:nbn:de:0030-drops-106487}, doi = {10.4230/LIPIcs.ICALP.2019.72}, annote = {Keywords: Balancing sets, depth-2 threshold circuits, polynomials, majority, weighted thresholds} }

Document

**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

The isoperimetric profile of a graph is a function that measures, for an integer k, the size of the smallest edge boundary over all sets of vertices of size k. We observe a connection between isoperimetric profiles and computational complexity. We illustrate this connection by an example from communication complexity, but our main result is in algebraic complexity.
We prove a sharp super-polynomial separation between monotone arithmetic circuits and monotone arithmetic branching programs. This shows that the classical simulation of arithmetic circuits by arithmetic branching programs by Valiant, Skyum, Berkowitz, and Rackoff (1983) cannot be improved, as long as it preserves monotonicity.
A key ingredient in the proof is an accurate analysis of the isoperimetric profile of finite full binary trees. We show that the isoperimetric profile of a full binary tree constantly fluctuates between one and almost the depth of the tree.

Pavel Hrubes and Amir Yehudayoff. On Isoperimetric Profiles and Computational Complexity. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 89:1-89:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{hrubes_et_al:LIPIcs.ICALP.2016.89, author = {Hrubes, Pavel and Yehudayoff, Amir}, title = {{On Isoperimetric Profiles and Computational Complexity}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {89:1--89:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.89}, URN = {urn:nbn:de:0030-drops-61964}, doi = {10.4230/LIPIcs.ICALP.2016.89}, annote = {Keywords: Monotone computation, separations, communication complexity, isoperimetry} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

In this paper, we compare the strength of the semantic and syntactic version of the cutting planes proof system.
First, we show that the lower bound technique of Pudlák applies also to semantic cutting planes: the proof system has feasible interpolation via monotone real circuits, which gives an exponential lower bound on lengths of semantic cutting planes refutations.
Second, we show that semantic refutations are stronger than syntactic ones. In particular, we give a formula for which any refutation in syntactic cutting planes requires exponential length, while there is a polynomial length refutation in semantic cutting planes. In other words, syntactic cutting planes does not p-simulate semantic cutting planes. We also give two incompatible integer inequalities which require exponential length refutation in syntactic cutting planes.
Finally, we pose the following problem, which arises in connection with semantic inference of arity larger than two: can every multivariate non-decreasing real function be expressed as a composition of non-decreasing real functions in two variables?

Yuval Filmus, Pavel Hrubeš, and Massimo Lauria. Semantic Versus Syntactic Cutting Planes. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{filmus_et_al:LIPIcs.STACS.2016.35, author = {Filmus, Yuval and Hrube\v{s}, Pavel and Lauria, Massimo}, title = {{Semantic Versus Syntactic Cutting Planes}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {35:1--35:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.35}, URN = {urn:nbn:de:0030-drops-57367}, doi = {10.4230/LIPIcs.STACS.2016.35}, annote = {Keywords: proof complexity, cutting planes, lower bounds} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

We consider boolean circuits in which every gate may compute an arbitrary boolean function of k other gates, for a parameter k. We give an explicit function $f:{0,1}^n -> {0,1} that requires at least Omega(log^2(n)) non-input gates when k = 2n/3. When the circuit is restricted to being layered and depth 2, we prove a lower bound of n^(Omega(1)) on the number of non-input gates. When the circuit is a formula with gates of fan-in k, we give a lower bound Omega(n^2/k*log(n)) on the total number of gates.
Our model is connected to some well known approaches to proving lower bounds in complexity theory. Optimal lower bounds for the Number-On-Forehead model in communication complexity, or for bounded depth circuits in AC_0, or extractors for varieties over small fields would imply strong lower bounds in our model. On the other hand, new lower bounds for our model would prove new time-space tradeoffs for branching programs and impossibility results for (fan-in 2) circuits with linear size and logarithmic depth. In particular, our lower bound gives a different proof for a known time-space tradeoff for oblivious branching programs.

Pavel Hrubes and Anup Rao. Circuits with Medium Fan-In. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 381-391, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hrubes_et_al:LIPIcs.CCC.2015.381, author = {Hrubes, Pavel and Rao, Anup}, title = {{Circuits with Medium Fan-In}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {381--391}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.381}, URN = {urn:nbn:de:0030-drops-50528}, doi = {10.4230/LIPIcs.CCC.2015.381}, annote = {Keywords: Boolean circuit, Complexity, Communication Complexity} }