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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-dev.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} }

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

We prove new cell-probe lower bounds for dynamic data structures that maintain a subset of {1,2,...,n}, and compute various statistics of the set. The data structure is said to handle insertions non-adaptively if the locations of memory accessed depend only on the element being inserted, and not on the contents of the memory. For any such data structure that can compute the median of the set, we prove that: t_{med} >= Omega(n^{1/(t_{ins}+1)}/(w^2 * t_{ins}^2)), where t_{ins} is the number of memory locations accessed during insertions, t_{med} is the number of memory locations accessed to compute the median, and w is the number of bits stored in each memory location. When the data structure is able to perform deletions non-adaptively and compute the minimum non-adaptively, we prove t_{min} + t_{del} >= Omega(log n /(log w + log log n)), where t_{min} is the number of locations accessed to compute the minimum, and t_{del} is the number of locations accessed to perform deletions. For the predecessor search problem, where the data structure is required to compute the predecessor of any element in the set, we prove that if computing the predecessors can be done non-adaptively, then either t_{pred} >= Omega(log n/(log log n + log w)), or t_{ins} >= Omega(n^{1/(2(t_{pred}+1))}), where t_{pred} is the number of locations accessed to compute predecessors.
These bounds are nearly matched by Binary Search Trees in some range of parameters. Our results follow from using the Sunflower Lemma of Erdös and Rado [Paul Erdös and Richard Rado, 1960] together with several kinds of encoding arguments.

Sivaramakrishnan Natarajan Ramamoorthy and Anup Rao. Lower Bounds on Non-Adaptive Data Structures Maintaining Sets of Numbers, from Sunflowers. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{natarajanramamoorthy_et_al:LIPIcs.CCC.2018.27, author = {Natarajan Ramamoorthy, Sivaramakrishnan and Rao, Anup}, title = {{Lower Bounds on Non-Adaptive Data Structures Maintaining Sets of Numbers, from Sunflowers}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {27:1--27:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.27}, URN = {urn:nbn:de:0030-drops-88625}, doi = {10.4230/LIPIcs.CCC.2018.27}, annote = {Keywords: Non-adaptive data structures, Sunflower lemma} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

We study a direct-sum question for read-once branching programs. If M(f) denotes the minimum average memory required to compute a function f(x_1,x_2, ..., x_n) how much memory is required to compute f on k independent inputs that arrive in parallel? We show that when the inputs are sampled independently from some domain X and M(f) = Omega(n), then computing the value of f on k streams requires average memory at least Omega(k * M(f)/n).
Our results are obtained by defining new ways to measure the information complexity of read-once branching programs. We define two such measures: the transitional and cumulative information content. We prove that any read-once branching program with transitional information content I can be simulated using average memory O(n(I+1)). On the other hand, if every read-once branching program with cumulative information content I can be simulated with average memory O(I+1), then computing f on k inputs requires average memory at least Omega(k * (M(f)-1)).

Anup Rao and Makrand Sinha. A Direct-Sum Theorem for Read-Once Branching Programs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{rao_et_al:LIPIcs.APPROX-RANDOM.2016.44, author = {Rao, Anup and Sinha, Makrand}, title = {{A Direct-Sum Theorem for Read-Once Branching Programs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, 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.2016.44}, URN = {urn:nbn:de:0030-drops-66676}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.44}, annote = {Keywords: Direct-sum, Information complexity, Streaming Algorithms} }

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Complete Volume

**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

LIPIcs, Volume 40, APPROX/RANDOM'15, Complete Volume

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Proceedings{garg_et_al:LIPIcs.APPROX-RANDOM.2015, title = {{LIPIcs, Volume 40, APPROX/RANDOM'15, Complete Volume}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, 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.2015}, URN = {urn:nbn:de:0030-drops-54012}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015}, annote = {Keywords: Data Structures, Coding and Information Theory, Theory of Computation, Computation by Abstract Devices, Modes of Computation, Complexity Measures and Problem Complexity, Numerical Algorithms and Problems, Nonnumerical Algorithms and Problems, Approximation, Numerical Linear Algorithms and Problems} }

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Front Matter

**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Frontmatter, Table of Contents, Preface, Program Commitees, External Reviewers, List of Authors

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. i-xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{garg_et_al:LIPIcs.APPROX-RANDOM.2015.i, author = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, title = {{Frontmatter, Table of Contents, Preface, Program Commitees, External Reviewers, List of Authors}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {i--xviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, 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.2015.i}, URN = {urn:nbn:de:0030-drops-53474}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.i}, annote = {Keywords: Frontmatter, Table of Contents, Preface, Program Commitees, External Reviewers, List of Authors} }

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

We show that the deterministic number-on-forehead communication complexity of set disjointness for k parties on a universe of size n is Omega(n/4^k). This gives the first lower bound that is linear in n, nearly matching Grolmusz's upper bound of O(log^2(n) + k^2n/2^k). We also simplify the proof of Sherstov's Omega(sqrt(n)/(k2^k)) lower bound for the randomized communication complexity of set disjointness.

Anup Rao and Amir Yehudayoff. Simplified Lower Bounds on the Multiparty Communication Complexity of Disjointness. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 88-101, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{rao_et_al:LIPIcs.CCC.2015.88, author = {Rao, Anup and Yehudayoff, Amir}, title = {{Simplified Lower Bounds on the Multiparty Communication Complexity of Disjointness}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {88--101}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.88}, URN = {urn:nbn:de:0030-drops-50769}, doi = {10.4230/LIPIcs.CCC.2015.88}, annote = {Keywords: communication complexity, set disjointness, number on forehead, lower bounds} }

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

We study the relationship between communication and information in 2-party communication protocols when the information is asymmetric. If I^A denotes the number of bits of information revealed by the first party, I^B denotes the information revealed by the second party, and C is the number of bits of communication in the protocol, we show that i) one can simulate the protocol using order I^A + (C^3 * I^B)^(1/4) * log(C) + (C * I^B)^(1/2) * log(C) bits of communication, ii) one can simulate the protocol using order I^A * 2^(O(I^B)) bits of communication The first result gives the best known bound on the complexity of a simulation when I^A >> I^B,C^(3/4). The second gives the best known bound when I^B << log C. In addition we show that if a function is computed by a protocol with asymmetric information complexity, then the inputs must have a large, nearly monochromatic rectangle of the right dimensions, a fact that is useful for proving lower bounds on lopsided communication problems.

Sivaramakrishnan Natarajan Ramamoorthy and Anup Rao. How to Compress Asymmetric Communication. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 102-123, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{natarajanramamoorthy_et_al:LIPIcs.CCC.2015.102, author = {Natarajan Ramamoorthy, Sivaramakrishnan and Rao, Anup}, title = {{How to Compress Asymmetric Communication}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {102--123}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.102}, URN = {urn:nbn:de:0030-drops-50679}, doi = {10.4230/LIPIcs.CCC.2015.102}, annote = {Keywords: Communication Complexity, Interactive Compression, Information Complexity} }

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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-dev.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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