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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We present efficient algorithms for solving systems of linear equations in 1-Laplacians of well-shaped simplicial complexes. 1-Laplacians, or higher-dimensional Laplacians, generalize graph Laplacians to higher-dimensional simplicial complexes and play a key role in computational topology and topological data analysis. Previously, nearly-linear time solvers were developed for simplicial complexes with known collapsing sequences and bounded Betti numbers, such as those triangulating a three-ball in ℝ³ (Cohen, Fasy, Miller, Nayyeri, Peng, and Walkington [SODA'2014], Black, Maxwell, Nayyeri, and Winkelman [SODA'2022], Black and Nayyeri [ICALP'2022]). Furthermore, Nested Dissection provides quadratic time solvers for more general systems with nonzero structures representing well-shaped simplicial complexes embedded in ℝ³.
We generalize the specialized solvers for 1-Laplacians to simplicial complexes with additional geometric structures but without collapsing sequences and bounded Betti numbers, and we improve the runtime of Nested Dissection. We focus on simplicial complexes that meet two conditions: (1) each individual simplex has a bounded aspect ratio, and (2) they can be divided into "disjoint" and balanced regions with well-shaped interiors and boundaries. Our solvers draw inspiration from the Incomplete Nested Dissection for stiffness matrices of well-shaped trusses (Kyng, Peng, Schwieterman, and Zhang [STOC'2018]).

Ming Ding and Peng Zhang. Efficient 1-Laplacian Solvers for Well-Shaped Simplicial Complexes: Beyond Betti Numbers and Collapsing Sequences. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 41:1-41:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ding_et_al:LIPIcs.ESA.2023.41, author = {Ding, Ming and Zhang, Peng}, title = {{Efficient 1-Laplacian Solvers for Well-Shaped Simplicial Complexes: Beyond Betti Numbers and Collapsing Sequences}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {41:1--41:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.41}, URN = {urn:nbn:de:0030-drops-186947}, doi = {10.4230/LIPIcs.ESA.2023.41}, annote = {Keywords: 1-Laplacian Solvers, Simplicial Complexes, Incomplete Nested Dissection} }

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APPROX

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Marcus, Spielman and Srivastava (Annals of Mathematics 2014) solved the Kadison-Singer Problem by proving a strong form of Weaver’s conjecture: they showed that for all α > 0 and all lists of vectors of norm at most √α whose outer products sum to the identity, there exists a signed sum of those outer products with operator norm at most √{8α} + 2α. We prove that it is NP-hard to distinguish such a list of vectors for which there is a signed sum that equals the zero matrix from those in which every signed sum has operator norm at least η √α, for some absolute constant η > 0. Thus, it is NP-hard to construct a signing that is a constant factor better than that guaranteed to exist.
For α = 1/4, we prove that it is NP-hard to distinguish whether there is a signed sum that equals the zero matrix from the case in which every signed sum has operator norm at least 1/4.

Daniel A. Spielman and Peng Zhang. Hardness Results for Weaver’s Discrepancy Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 40:1-40:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{spielman_et_al:LIPIcs.APPROX/RANDOM.2022.40, author = {Spielman, Daniel A. and Zhang, Peng}, title = {{Hardness Results for Weaver’s Discrepancy Problem}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {40:1--40:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.40}, URN = {urn:nbn:de:0030-drops-171628}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.40}, annote = {Keywords: Discrepancy Problem, Kadison-Singer Problem, Hardness of Approximation} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We study linear equations in combinatorial Laplacians of k-dimensional simplicial complexes (k-complexes), a natural generalization of graph Laplacians. Combinatorial Laplacians play a crucial role in homology and are a central tool in topology. Beyond this, they have various applications in data analysis and physical modeling problems. It is known that nearly-linear time solvers exist for graph Laplacians. However, nearly-linear time solvers for combinatorial Laplacians are only known for restricted classes of complexes.
This paper shows that linear equations in combinatorial Laplacians of 2-complexes are as hard to solve as general linear equations. More precisely, for any constant c ≥ 1, if we can solve linear equations in combinatorial Laplacians of 2-complexes up to high accuracy in time Õ((# of nonzero coefficients)^c), then we can solve general linear equations with polynomially bounded integer coefficients and condition numbers up to high accuracy in time Õ((# of nonzero coefficients)^c). We prove this by a nearly-linear time reduction from general linear equations to combinatorial Laplacians of 2-complexes. Our reduction preserves the sparsity of the problem instances up to poly-logarithmic factors.

Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, and Peng Zhang. Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 53:1-53:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ding_et_al:LIPIcs.ICALP.2022.53, author = {Ding, Ming and Kyng, Rasmus and Gutenberg, Maximilian Probst and Zhang, Peng}, title = {{Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {53:1--53:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.53}, URN = {urn:nbn:de:0030-drops-163945}, doi = {10.4230/LIPIcs.ICALP.2022.53}, annote = {Keywords: Simplicial Complexes, Combinatorial Laplacians, Linear Equations, Fine-Grained Complexity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We give a nearly-linear time reduction that encodes any linear program as a 2-commodity flow problem with only a small blow-up in size. Under mild assumptions similar to those employed by modern fast solvers for linear programs, our reduction causes only a polylogarithmic multiplicative increase in the size of the program, and runs in nearly-linear time. Our reduction applies to high-accuracy approximation algorithms and exact algorithms. Given an approximate solution to the 2-commodity flow problem, we can extract a solution to the linear program in linear time with only a polynomial factor increase in the error. This implies that any algorithm that solves the 2-commodity flow problem can solve linear programs in essentially the same time. Given a directed graph with edge capacities and two source-sink pairs, the goal of the 2-commodity flow problem is to maximize the sum of the flows routed between the two source-sink pairs subject to edge capacities and flow conservation. A 2-commodity flow problem can be formulated as a linear program, which can be solved to high accuracy in almost the current matrix multiplication time (Cohen-Lee-Song JACM'21). Our reduction shows that linear programs can be approximately solved, to high accuracy, using 2-commodity flow as well.
Our proof follows the outline of Itai’s polynomial-time reduction of a linear program to a 2-commodity flow problem (JACM’78). Itai’s reduction shows that exactly solving 2-commodity flow and exactly solving linear programming are polynomial-time equivalent. We improve Itai’s reduction to nearly preserve the problem representation size in each step. In addition, we establish an error bound for approximately solving each intermediate problem in the reduction, and show that the accumulated error is polynomially bounded. We remark that our reduction does not run in strongly polynomial time and that it is open whether 2-commodity flow and linear programming are equivalent in strongly polynomial time.

Ming Ding, Rasmus Kyng, and Peng Zhang. Two-Commodity Flow Is Equivalent to Linear Programming Under Nearly-Linear Time Reductions. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 54:1-54:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ding_et_al:LIPIcs.ICALP.2022.54, author = {Ding, Ming and Kyng, Rasmus and Zhang, Peng}, title = {{Two-Commodity Flow Is Equivalent to Linear Programming Under Nearly-Linear Time Reductions}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {54:1--54:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.54}, URN = {urn:nbn:de:0030-drops-163950}, doi = {10.4230/LIPIcs.ICALP.2022.54}, annote = {Keywords: Two-Commodity Flow Problems, Linear Programming, Fine-Grained Complexity} }

Document

**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

The maximum duo-preservation string mapping (Max-Duo) problem is
the complement of the well studied minimum common string partition (MCSP) problem, both of which have applications in many fields including text compression and bioinformatics. k-Max-Duo is the restricted version of Max-Duo, where every letter of the alphabet occurs at most k times in each of the strings, which is readily reduced into the well known maximum independent set (MIS) problem on a graph of maximum degree \Delta \le 6(k-1). In particular, 2-Max-Duo can then be approximated arbitrarily close to 1.8 using the state-of-the-art approximation algorithm for the MIS problem. 2-Max-Duo was proved APX-hard and very recently a (1.6 + \epsilon)-approximation was claimed, for any \epsilon > 0. In this paper, we present a vertex-degree reduction technique, based on which, we show that 2-Max-Duo can be approximated arbitrarily close to 1.4.

Yao Xu, Yong Chen, Guohui Lin, Tian Liu, Taibo Luo, and Peng Zhang. A (1.4 + epsilon)-Approximation Algorithm for the 2-Max-Duo Problem. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 66:1-66:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{xu_et_al:LIPIcs.ISAAC.2017.66, author = {Xu, Yao and Chen, Yong and Lin, Guohui and Liu, Tian and Luo, Taibo and Zhang, Peng}, title = {{A (1.4 + epsilon)-Approximation Algorithm for the 2-Max-Duo Problem}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {66:1--66:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.66}, URN = {urn:nbn:de:0030-drops-82120}, doi = {10.4230/LIPIcs.ISAAC.2017.66}, annote = {Keywords: Approximation algorithm, duo-preservation string mapping, string partition, independent set} }

Document

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

We study the problem of approximately solving positive linear programs (LPs). This class of LPs models a wide range of fundamental problems in combinatorial optimization and operations research, such as many resource allocation problems, solving non-negative linear systems, computing tomography, single/multi commodity flows on graphs, etc. For the special cases of pure packing or pure covering LPs, recent result by Allen-Zhu and Orecchia [Allen/Zhu/Orecchia, SODA'15] gives O˜(1/(epsilon^3))-time parallel algorithm, which breaks the longstanding O˜(1/(epsilon^4)) running time bound by the seminal work of Luby and Nisan [Luby/Nisan, STOC'93].
We present new parallel algorithm with running time O˜(1/(epsilon^3)) for the more general mixed packing and covering LPs, which improves upon the O˜(1/(epsilon^4))-time algorithm of Young [Young, FOCS'01; Young, arXiv 2014]. Our work leverages the ideas from both the optimization oriented approach [Allen/Zhu/Orecchia, SODA'15; Wang/Mahoney/Mohan/Rao, arXiv 2015], as well as the more combinatorial approach with phases [Young, FOCS'01; Young, arXiv 2014]. In addition, our algorithm, when directly applied to pure packing or pure covering LPs, gives a improved running time of O˜(1/(epsilon^2)).

Michael W. Mahoney, Satish Rao, Di Wang, and Peng Zhang. Approximating the Solution to Mixed Packing and Covering LPs in Parallel O˜(epsilon^{-3}) Time. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 52:1-52:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mahoney_et_al:LIPIcs.ICALP.2016.52, author = {Mahoney, Michael W. and Rao, Satish and Wang, Di and Zhang, Peng}, title = {{Approximating the Solution to Mixed Packing and Covering LPs in Parallel O˜(epsilon^\{-3\}) Time}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {52:1--52:14}, 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.2016.52}, URN = {urn:nbn:de:0030-drops-63335}, doi = {10.4230/LIPIcs.ICALP.2016.52}, annote = {Keywords: Mixed packing and covering, Linear program, Approximation algorithm, Parallel algorithm} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 8372, Computer Science in Sport - Mission and Methods (2008)

Our research focuses on learning approaches with robot KiRo. KiRo is
a table soccer robot which can challenge even advanced human
players. Previously, we developed a method using learning by imitation,
by which KiRo can automatically acquire the demonstrated actions. Recently, we
constructed a game-recorder which collects data from the human-played games.
The in-process work is about explaining the recorded data, which is to classify and
to evaluate human's skills. A brief overview of the previous work is addressed,
and the perspective is discussed.

Dapeng Zhang. Learning with Table Soccer. In Computer Science in Sport - Mission and Methods. Dagstuhl Seminar Proceedings, Volume 8372, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{zhang:DagSemProc.08372.6, author = {Zhang, Dapeng}, title = {{Learning with Table Soccer}}, booktitle = {Computer Science in Sport - Mission and Methods}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8372}, editor = {Arnold Baca and Martin Lames and Keith Lyons and Bernhard Nebel and Josef Wiemeyer}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08372.6}, URN = {urn:nbn:de:0030-drops-16839}, doi = {10.4230/DagSemProc.08372.6}, annote = {Keywords: Table Soccer Robot, Learning} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

A folklore conjecture in quantum computing is that the acceptance probability of a quantum query algorithm can be approximated by a classical decision tree, with only a polynomial increase in the number of queries. Motivated by this conjecture, Aaronson and Ambainis (Theory of Computing, 2014) conjectured that this should hold more generally for any bounded function computed by a low degree polynomial.
In this work we prove two new results towards establishing this conjecture: first, that any such polynomial has a small fractional certificate complexity; and second, that many inputs have a small sensitive block. We show that these would imply the Aaronson and Ambainis conjecture, assuming a conjectured extension of Talagrand’s concentration inequality.
On the technical side, many classical techniques used in the analysis of Boolean functions seem to fail when applied to bounded functions. Here, we develop a new technique, based on a mix of combinatorics, analysis and geometry, and which in part extends a recent technique of Knop et al. (STOC 2021) to bounded functions.

Shachar Lovett and Jiapeng Zhang. Fractional Certificates for Bounded Functions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 84:1-84:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lovett_et_al:LIPIcs.ITCS.2023.84, author = {Lovett, Shachar and Zhang, Jiapeng}, title = {{Fractional Certificates for Bounded Functions}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {84:1--84:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.84}, URN = {urn:nbn:de:0030-drops-175871}, doi = {10.4230/LIPIcs.ITCS.2023.84}, annote = {Keywords: Aaronson-Ambainis conjecture, fractional block sensitivity, Talagrand inequality} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Query-to-communication lifting theorems translate lower bounds on query complexity to lower bounds for the corresponding communication model. In this paper, we give a simplified proof of deterministic lifting (in both the tree-like and dag-like settings). Our proof uses elementary counting together with a novel connection to the sunflower lemma.
In addition to a simplified proof, our approach opens up a new avenue of attack towards proving lifting theorems with improved gadget size - one of the main challenges in the area. Focusing on one of the most widely used gadgets - the index gadget - existing lifting techniques are known to require at least a quadratic gadget size. Our new approach combined with robust sunflower lemmas allows us to reduce the gadget size to near linear. We conjecture that it can be further improved to polylogarithmic, similar to the known bounds for the corresponding robust sunflower lemmas.

Shachar Lovett, Raghu Meka, Ian Mertz, Toniann Pitassi, and Jiapeng Zhang. Lifting with Sunflowers. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 104:1-104:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lovett_et_al:LIPIcs.ITCS.2022.104, author = {Lovett, Shachar and Meka, Raghu and Mertz, Ian and Pitassi, Toniann and Zhang, Jiapeng}, title = {{Lifting with Sunflowers}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {104:1--104:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.104}, URN = {urn:nbn:de:0030-drops-157004}, doi = {10.4230/LIPIcs.ITCS.2022.104}, annote = {Keywords: Lifting theorems, communication complexity, combinatorics, sunflowers} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

The sunflower conjecture is one of the most well-known open problems in combinatorics. It has several applications in theoretical computer science, one of which is DNF compression, due to Gopalan, Meka and Reingold (Computational Complexity, 2013). In this paper, we show that improved bounds for DNF compression imply improved bounds for the sunflower conjecture, which is the reverse direction of the DNF compression result. The main approach is based on regularity of set systems and a structure-vs-pseudorandomness approach to the sunflower conjecture.

Shachar Lovett, Noam Solomon, and Jiapeng Zhang. From DNF Compression to Sunflower Theorems via Regularity. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{lovett_et_al:LIPIcs.CCC.2019.5, author = {Lovett, Shachar and Solomon, Noam and Zhang, Jiapeng}, title = {{From DNF Compression to Sunflower Theorems via Regularity}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.5}, URN = {urn:nbn:de:0030-drops-108277}, doi = {10.4230/LIPIcs.CCC.2019.5}, annote = {Keywords: DNF sparsification, sunflower conjecture, regular set systems} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

The Erdös-Rado sunflower theorem (Journal of Lond. Math. Soc. 1960) is a fundamental result in combinatorics, and the corresponding sunflower conjecture is a central open problem. Motivated by applications in complexity theory, Rossman (FOCS 2010) extended the result to quasi-sunflowers, where similar conjectures emerge about the optimal parameters for which it holds.
In this work, we exhibit a surprising connection between the existence of sunflowers and quasi-sunflowers in large enough set systems, and the problem of constructing (or existing) certain randomness extractors. This allows us to re-derive the known results in a systematic manner, and to reduce the relevant conjectures to the problem of obtaining improved constructions of the randomness extractors.

Xin Li, Shachar Lovett, and Jiapeng Zhang. Sunflowers and Quasi-Sunflowers from Randomness Extractors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 51:1-51:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{li_et_al:LIPIcs.APPROX-RANDOM.2018.51, author = {Li, Xin and Lovett, Shachar and Zhang, Jiapeng}, title = {{Sunflowers and Quasi-Sunflowers from Randomness Extractors}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {51:1--51:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.51}, URN = {urn:nbn:de:0030-drops-94555}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.51}, annote = {Keywords: Sunflower conjecture, Quasi-sunflowers, Randomness Extractors} }

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

We study entropy flattening: Given a circuit C_X implicitly describing an n-bit source X (namely, X is the output of C_X on a uniform random input), construct another circuit C_Y describing a source Y such that (1) source Y is nearly flat (uniform on its support), and (2) the Shannon entropy of Y is monotonically related to that of X. The standard solution is to have C_Y evaluate C_X altogether Theta(n^2) times on independent inputs and concatenate the results (correctness follows from the asymptotic equipartition property). In this paper, we show that this is optimal among black-box constructions: Any circuit C_Y for entropy flattening that repeatedly queries C_X as an oracle requires Omega(n^2) queries.
Entropy flattening is a component used in the constructions of pseudorandom generators and other cryptographic primitives from one-way functions [Johan Håstad et al., 1999; John Rompel, 1990; Thomas Holenstein, 2006; Iftach Haitner et al., 2006; Iftach Haitner et al., 2009; Iftach Haitner et al., 2013; Iftach Haitner et al., 2010; Salil P. Vadhan and Colin Jia Zheng, 2012]. It is also used in reductions between problems complete for statistical zero-knowledge [Tatsuaki Okamoto, 2000; Amit Sahai and Salil P. Vadhan, 1997; Oded Goldreich et al., 1999; Vadhan, 1999]. The Theta(n^2) query complexity is often the main efficiency bottleneck. Our lower bound can be viewed as a step towards proving that the current best construction of pseudorandom generator from arbitrary one-way functions by Vadhan and Zheng (STOC 2012) has optimal efficiency.

Yi-Hsiu Chen, Mika Göös, Salil P. Vadhan, and Jiapeng Zhang. A Tight Lower Bound for Entropy Flattening. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 23:1-23:28, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chen_et_al:LIPIcs.CCC.2018.23, author = {Chen, Yi-Hsiu and G\"{o}\"{o}s, Mika and Vadhan, Salil P. and Zhang, Jiapeng}, title = {{A Tight Lower Bound for Entropy Flattening}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {23:1--23:28}, 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.2018.23}, URN = {urn:nbn:de:0030-drops-88669}, doi = {10.4230/LIPIcs.CCC.2018.23}, annote = {Keywords: Entropy, One-way function} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Classical results in computability theory, notably Rice’s theorem, focus on the extensional content of programs, namely, on the partial recursive functions that programs compute. Later and more recent work investigated intensional generalisations of such results that take into account the way in which functions are computed, thus affected by the specific programs computing them. In this paper, we single out a novel class of program semantics based on abstract domains of program properties that are able to capture nonextensional aspects of program computations, such as their asymptotic complexity or logical invariants, and allow us to generalise some foundational computability results such as Rice’s Theorem and Kleene’s Second Recursion Theorem to these semantics. In particular, it turns out that for this class of abstract program semantics, any nontrivial abstract property is undecidable and every decidable overapproximation necessarily includes an infinite set of false positives which covers all values of the semantic abstract domain.

Paolo Baldan, Francesco Ranzato, and Linpeng Zhang. A Rice’s Theorem for Abstract Semantics. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 117:1-117:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{baldan_et_al:LIPIcs.ICALP.2021.117, author = {Baldan, Paolo and Ranzato, Francesco and Zhang, Linpeng}, title = {{A Rice’s Theorem for Abstract Semantics}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {117:1--117:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.117}, URN = {urn:nbn:de:0030-drops-141860}, doi = {10.4230/LIPIcs.ICALP.2021.117}, annote = {Keywords: Computability Theory, Recursive Function, Rice’s Theorem, Kleene’s Second Recursion Theorem, Program Analysis, Affine Program Invariants} }

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