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

Let G = (V,w) be a weighted undirected graph with m edges. The cut dimension of G is the dimension of the span of the characteristic vectors of the minimum cuts of G, viewed as vectors in {0,1}^m. For every n ≥ 2 we show that the cut dimension of an n-vertex graph is at most 2n-3, and construct graphs realizing this bound.
The cut dimension was recently defined by Graur et al. [Andrei Graur et al., 2020], who show that the maximum cut dimension of an n-vertex graph is a lower bound on the number of cut queries needed by a deterministic algorithm to solve the minimum cut problem on n-vertex graphs. For every n ≥ 2, Graur et al. exhibit a graph on n vertices with cut dimension at least 3n/2 -2, giving the first lower bound larger than n on the deterministic cut query complexity of computing mincut. We observe that the cut dimension is even a lower bound on the number of linear queries needed by a deterministic algorithm to solve mincut, where a linear query can ask any vector x ∈ ℝ^{binom(n,2)} and receives the answer w^T x. Our results thus show a lower bound of 2n-3 on the number of linear queries needed by a deterministic algorithm to solve minimum cut on n-vertex graphs, and imply that one cannot show a lower bound larger than this via the cut dimension.
We further introduce a generalization of the cut dimension which we call the 𝓁₁-approximate cut dimension. The 𝓁₁-approximate cut dimension is also a lower bound on the number of linear queries needed by a deterministic algorithm to compute minimum cut. It is always at least as large as the cut dimension, and we construct an infinite family of graphs on n = 3k+1 vertices with 𝓁₁-approximate cut dimension 2n-2, showing that it can be strictly larger than the cut dimension.

Troy Lee, Tongyang Li, Miklos Santha, and Shengyu Zhang. On the Cut Dimension of a Graph. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 15:1-15:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{lee_et_al:LIPIcs.CCC.2021.15, author = {Lee, Troy and Li, Tongyang and Santha, Miklos and Zhang, Shengyu}, title = {{On the Cut Dimension of a Graph}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {15:1--15:35}, 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.2021.15}, URN = {urn:nbn:de:0030-drops-142890}, doi = {10.4230/LIPIcs.CCC.2021.15}, annote = {Keywords: Query complexity, submodular function minimization, cut dimension} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for f(x and y) and f(x xor y) with monotone functions f, where and and xor are bit-wise AND and XOR , respectively. In this paper, we extend these results to functions f which alternate values for a relatively small number of times on any monotone path from 0^n to 1^n. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers.

Chengyu Lin and Shengyu Zhang. Sensitivity Conjecture and Log-Rank Conjecture for Functions with Small Alternating Numbers. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{lin_et_al:LIPIcs.ICALP.2017.51, author = {Lin, Chengyu and Zhang, Shengyu}, title = {{Sensitivity Conjecture and Log-Rank Conjecture for Functions with Small Alternating Numbers}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {51:1--51:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.51}, URN = {urn:nbn:de:0030-drops-74045}, doi = {10.4230/LIPIcs.ICALP.2017.51}, annote = {Keywords: Analysis of Boolean functions, Sensitivity Conjecture, Log-rank Conjecture, Alternating Number} }

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

A canonical result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors Q_{ij} on a system of n qubits, and the task is to decide whether the Hamiltonian H = sum Q_{ij} has a 0-eigenvalue, or it is larger than 1/n^c for some c = O(1). The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding the ground state of 2-local frustration-free Hamiltonians of spin 1/2, a well-studied model believed to capture certain key properties in modern condensed matter physics. While Bravyi has shown that the quantum 2-SAT problem has a classical polynomial-time algorithm, the running time of his algorithm is O(n^4). In this paper we give a classical algorithm with linear running time in the number of local projectors, therefore achieving the best possible complexity.

Itai Arad, Miklos Santha, Aarthi Sundaram, and Shengyu Zhang. Linear Time Algorithm for Quantum 2SAT. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arad_et_al:LIPIcs.ICALP.2016.15, author = {Arad, Itai and Santha, Miklos and Sundaram, Aarthi and Zhang, Shengyu}, title = {{Linear Time Algorithm for Quantum 2SAT}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {15:1--15:14}, 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.2016.15}, URN = {urn:nbn:de:0030-drops-62795}, doi = {10.4230/LIPIcs.ICALP.2016.15}, annote = {Keywords: Quantum SAT, Davis-Putnam Procedure, Linear Time Algorithm} }

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

What is the minimum amount of information and time needed to solve 2SAT? When the instance is known, it can be solved in polynomial time, but is this also possible without knowing the instance? Bei, Chen and Zhang (STOC'13) considered a model where the input is accessed by proposing possible assignments to a special oracle. This oracle, on encountering some constraint unsatisfied by the proposal, returns only the constraint index. It turns out that, in this model, even 1SAT cannot be solved in polynomial time unless P=NP. Hence, we consider a model in which the input is accessed by proposing probability distributions over assignments to the variables. The oracle then returns the index of the constraint that is most likely to be violated by this distribution. We show that the information obtained this way is sufficient to solve 1SAT in polynomial time, even when the clauses can be repeated. For 2SAT, as long as there are no repeated clauses, in polynomial time we can even learn an equivalent formula for the hidden instance and hence also solve it. Furthermore, we extend these results to the quantum regime. We show that in this setting 1QSAT can be solved in polynomial time up to constant precision, and 2QSAT can be learnt in polynomial time up to inverse polynomial precision.

Itai Arad, Adam Bouland, Daniel Grier, Miklos Santha, Aarthi Sundaram, and Shengyu Zhang. On the Complexity of Probabilistic Trials for Hidden Satisfiability Problems. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arad_et_al:LIPIcs.MFCS.2016.12, author = {Arad, Itai and Bouland, Adam and Grier, Daniel and Santha, Miklos and Sundaram, Aarthi and Zhang, Shengyu}, title = {{On the Complexity of Probabilistic Trials for Hidden Satisfiability Problems}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {12:1--12:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.12}, URN = {urn:nbn:de:0030-drops-64284}, doi = {10.4230/LIPIcs.MFCS.2016.12}, annote = {Keywords: computational complexity, satisfiability problems, trial and error, quantum computing, learning theory} }

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**Published in:** LIPIcs, Volume 31, 18th International Conference on Database Theory (ICDT 2015)

In dynamic distinct counting, we want to maintain a multi-set S of integers under insertions to answer efficiently the query: how many distinct elements are there in S? In external memory, the problem admits two standard solutions. The first one maintains $S$ in a hash structure, so that the distinct count can be incrementally updated after each insertion using O(1) expected I/Os. A query is answered for free. The second one stores S in a linked list, and thus supports an insertion in O(1/B) amortized I/Os. A query can be answered in O(N/B log_{M/B} (N/B)) I/Os by sorting, where N=|S|, B is the block size, and M is the memory size.
In this paper, we show that the above two naive solutions are already optimal within a polylog factor. Specifically, for any Las Vegas structure using N^{O(1)} blocks, if its expected amortized insertion cost is o(1/log B}), then it must incur Omega(N/(B log B)) expected I/Os answering a query in the worst case, under the (realistic) condition that N is a polynomial of B. This means that the problem is repugnant to update buffering: the query cost jumps from 0 dramatically to almost linearity as soon as the insertion cost drops slightly below Omega(1).

Xiaocheng Hu, Yufei Tao, Yi Yang, Shengyu Zhang, and Shuigeng Zhou. On The I/O Complexity of Dynamic Distinct Counting. In 18th International Conference on Database Theory (ICDT 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 31, pp. 265-276, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hu_et_al:LIPIcs.ICDT.2015.265, author = {Hu, Xiaocheng and Tao, Yufei and Yang, Yi and Zhang, Shengyu and Zhou, Shuigeng}, title = {{On The I/O Complexity of Dynamic Distinct Counting}}, booktitle = {18th International Conference on Database Theory (ICDT 2015)}, pages = {265--276}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-79-8}, ISSN = {1868-8969}, year = {2015}, volume = {31}, editor = {Arenas, Marcelo and Ugarte, Mart{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2015.265}, URN = {urn:nbn:de:0030-drops-49895}, doi = {10.4230/LIPIcs.ICDT.2015.265}, annote = {Keywords: distinct counting, lower bound, external memory} }

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