19 Search Results for "Sun, He"

Document
DOI: 10.4230/LIPIcs.CCC.2024.7
Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy

Authors: Sepehr Assadi, Prantar Ghosh, Bruno Loff, Parth Mittal, and Sagnik Mukhopadhyay

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

Abstract
The following question arises naturally in the study of graph streaming algorithms: Is there any graph problem which is "not too hard", in that it can be solved efficiently with total communication (nearly) linear in the number n of vertices, and for which, nonetheless, any streaming algorithm with Õ(n) space (i.e., a semi-streaming algorithm) needs a polynomial n^Ω(1) number of passes? Assadi, Chen, and Khanna [STOC 2019] were the first to prove that this is indeed the case. However, the lower bounds that they obtained are for rather non-standard graph problems. Our first main contribution is to present the first polynomial-pass lower bounds for natural "not too hard" graph problems studied previously in the streaming model: k-cores and degeneracy. We devise a novel communication protocol for both problems with near-linear communication, thus showing that k-cores and degeneracy are natural examples of "not too hard" problems. Indeed, previous work have developed single-pass semi-streaming algorithms for approximating these problems. In contrast, we prove that any semi-streaming algorithm for exactly solving these problems requires (almost) Ω(n^{1/3}) passes. The lower bound follows by a reduction from a generalization of the hidden pointer chasing (HPC) problem of Assadi, Chen, and Khanna, which is also the basis of their earlier semi-streaming lower bounds. Our second main contribution is improved round-communication lower bounds for the underlying communication problems at the basis of these reductions: - We improve the previous lower bound of Assadi, Chen, and Khanna for HPC to achieve optimal bounds for this problem. - We further observe that all current reductions from HPC can also work with a generalized version of this problem that we call MultiHPC, and prove an even stronger and optimal lower bound for this generalization. These two results collectively allow us to improve the resulting pass lower bounds for semi-streaming algorithms by a polynomial factor, namely, from n^{1/5} to n^{1/3} passes.
Cite as

Sepehr Assadi, Prantar Ghosh, Bruno Loff, Parth Mittal, and Sagnik Mukhopadhyay. Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{assadi_et_al:LIPIcs.CCC.2024.7,
  author =	{Assadi, Sepehr and Ghosh, Prantar and Loff, Bruno and Mittal, Parth and Mukhopadhyay, Sagnik},
  title =	{{Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{7:1--7:16},
  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.2024.7},
  URN =		{urn:nbn:de:0030-drops-204035},
  doi =		{10.4230/LIPIcs.CCC.2024.7},
  annote =	{Keywords: Graph streaming, Lower bounds, Communication complexity, k-Cores and degeneracy}
}
Document
DOI: 10.4230/LIPIcs.SEA.2024.27
Improved Cut Strategy for Tensor Network Contraction Orders

Authors: Christoph Staudt, Mark Blacher, Julien Klaus, Farin Lippmann, and Joachim Giesen

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
In the field of quantum computing, simulating quantum systems on classical computers is crucial. Tensor networks are fundamental in simulating quantum systems. A tensor network is a collection of tensors, that need to be contracted into a result tensor. Tensor contraction is a generalization of matrix multiplication to higher order tensors. The contractions can be performed in different orders, and the order has a significant impact on the number of floating point operations (flops) needed to get the result tensor. It is known that finding an optimal contraction order is NP-hard. The current state-of-the-art approach for finding efficient contraction orders is to combinine graph partitioning with a greedy strategy. Although heavily used in practice, the current approach ignores so-called free indices, chooses node weights without regarding previous computations, and requires numerous hyperparameters that need to be tuned at runtime. In this paper, we address these shortcomings by developing a novel graph cut strategy. The proposed modifications yield contraction orders that significantly reduce the number of flops in the tensor contractions compared to the current state of the art. Moreover, by removing the need for hyperparameter tuning at runtime, our approach converges to an efficient solution faster, which reduces the required optimization time by at least an order of magnitude.
Cite as

Christoph Staudt, Mark Blacher, Julien Klaus, Farin Lippmann, and Joachim Giesen. Improved Cut Strategy for Tensor Network Contraction Orders. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{staudt_et_al:LIPIcs.SEA.2024.27,
  author =	{Staudt, Christoph and Blacher, Mark and Klaus, Julien and Lippmann, Farin and Giesen, Joachim},
  title =	{{Improved Cut Strategy for Tensor Network Contraction Orders}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{27:1--27:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.27},
  URN =		{urn:nbn:de:0030-drops-203924},
  doi =		{10.4230/LIPIcs.SEA.2024.27},
  annote =	{Keywords: tensor network, contraction order, graph partitioniong, quantum simulation}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.11
Approximate Counting for Spin Systems in Sub-Quadratic Time

Authors: Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, and Jiaheng Wang

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We present two randomised approximate counting algorithms with Õ(n^{2-c}/ε²) running time for some constant c > 0 and accuracy ε: 1) for the hard-core model with fugacity λ on graphs with maximum degree Δ when λ = O(Δ^{-1.5-c₁}) where c₁ = c/(2-2c); 2) for spin systems with strong spatial mixing (SSM) on planar graphs with quadratic growth, such as ℤ². For the hard-core model, Weitz’s algorithm (STOC, 2006) achieves sub-quadratic running time when correlation decays faster than the neighbourhood growth, namely when λ = o(Δ^{-2}). Our first algorithm does not require this property and extends the range where sub-quadratic algorithms exist. Our second algorithm appears to be the first to achieve sub-quadratic running time up to the SSM threshold, albeit on a restricted family of graphs. It also extends to (not necessarily planar) graphs with polynomial growth, such as ℤ^d, but with a running time of the form Õ(n²ε^{-2}/2^{c(log n)^{1/d}}) where d is the exponent of the polynomial growth and c > 0 is some constant.
Cite as

Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, and Jiaheng Wang. Approximate Counting for Spin Systems in Sub-Quadratic Time. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{anand_et_al:LIPIcs.ICALP.2024.11,
  author =	{Anand, Konrad and Feng, Weiming and Freifeld, Graham and Guo, Heng and Wang, Jiaheng},
  title =	{{Approximate Counting for Spin Systems in Sub-Quadratic Time}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{11:1--11:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.11},
  URN =		{urn:nbn:de:0030-drops-201543},
  doi =		{10.4230/LIPIcs.ICALP.2024.11},
  annote =	{Keywords: Randomised algorithm, Approximate counting, Spin system, Sub-quadratic algorithm}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.37
Fast Approximate Counting of Cycles

Authors: Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a (1±ε)-approximation in Õ(n^ω/t^{ω-2}) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n× n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM(n,n/t^{1/(h-2)},n)), the time to multiply n × n/(t^{1/(h-2)}) by n/(t^{1/(h-2)) × n matrices. Finally, we show that under popular fine-grained hypotheses, this running time is optimal.
Cite as

Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams. Fast Approximate Counting of Cycles. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{censorhillel_et_al:LIPIcs.ICALP.2024.37,
  author =	{Censor-Hillel, Keren and Even, Tomer and Vassilevska Williams, Virginia},
  title =	{{Fast Approximate Counting of Cycles}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{37:1--37:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.37},
  URN =		{urn:nbn:de:0030-drops-201809},
  doi =		{10.4230/LIPIcs.ICALP.2024.37},
  annote =	{Keywords: Approximate triangle counting, Approximate cycle counting Fast matrix multiplication, Fast rectangular matrix multiplication}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.50
Fully-Scalable MPC Algorithms for Clustering in High Dimension

Authors: Artur Czumaj, Guichen Gao, Shaofeng H.-C. Jiang, Robert Krauthgamer, and Pavel Veselý

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We design new parallel algorithms for clustering in high-dimensional Euclidean spaces. These algorithms run in the Massively Parallel Computation (MPC) model, and are fully scalable, meaning that the local memory in each machine may be n^σ for arbitrarily small fixed σ > 0. Importantly, the local memory may be substantially smaller than the number of clusters k, yet all our algorithms are fast, i.e., run in O(1) rounds. We first devise a fast MPC algorithm for O(1)-approximation of uniform Facility Location. This is the first fully-scalable MPC algorithm that achieves O(1)-approximation for any clustering problem in general geometric setting; previous algorithms only provide poly(log n)-approximation or apply to restricted inputs, like low dimension or small number of clusters k; e.g. [Bhaskara and Wijewardena, ICML'18; Cohen-Addad et al., NeurIPS'21; Cohen-Addad et al., ICML'22]. We then build on this Facility Location result and devise a fast MPC algorithm that achieves O(1)-bicriteria approximation for k-Median and for k-Means, namely, it computes (1+ε)k clusters of cost within O(1/ε²)-factor of the optimum for k clusters. A primary technical tool that we introduce, and may be of independent interest, is a new MPC primitive for geometric aggregation, namely, computing for every data point a statistic of its approximate neighborhood, for statistics like range counting and nearest-neighbor search. Our implementation of this primitive works in high dimension, and is based on consistent hashing (aka sparse partition), a technique that was recently used for streaming algorithms [Czumaj et al., FOCS'22].
Cite as

Artur Czumaj, Guichen Gao, Shaofeng H.-C. Jiang, Robert Krauthgamer, and Pavel Veselý. Fully-Scalable MPC Algorithms for Clustering in High Dimension. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{czumaj_et_al:LIPIcs.ICALP.2024.50,
  author =	{Czumaj, Artur and Gao, Guichen and Jiang, Shaofeng H.-C. and Krauthgamer, Robert and Vesel\'{y}, Pavel},
  title =	{{Fully-Scalable MPC Algorithms for Clustering in High Dimension}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{50:1--50:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.50},
  URN =		{urn:nbn:de:0030-drops-201938},
  doi =		{10.4230/LIPIcs.ICALP.2024.50},
  annote =	{Keywords: Massively parallel computing, high dimension, facility location, k-median, k-means}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.119
Better Sparsifiers for Directed Eulerian Graphs

Authors: Sushant Sachdeva, Anvith Thudi, and Yibin Zhao

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Spectral sparsification for directed Eulerian graphs is a key component in the design of fast algorithms for solving directed Laplacian linear systems. Directed Laplacian linear system solvers are crucial algorithmic primitives to fast computation of fundamental problems on random walks, such as computing stationary distributions, hitting and commute times, and personalized PageRank vectors. While spectral sparsification is well understood for undirected graphs and it is known that for every graph G, (1+ε)-sparsifiers with O(nε^{-2}) edges exist [Batson-Spielman-Srivastava, STOC '09] (which is optimal), the best known constructions of Eulerian sparsifiers require Ω(nε^{-2}log⁴ n) edges and are based on short-cycle decompositions [Chu et al., FOCS '18]. In this paper, we give improved constructions of Eulerian sparsifiers, specifically: 1) We show that for every directed Eulerian graph G→, there exists an Eulerian sparsifier with O(nε^{-2} log² n log²log n + nε^{-4/3}log^{8/3} n) edges. This result is based on combining short-cycle decompositions [Chu-Gao-Peng-Sachdeva-Sawlani-Wang, FOCS '18, SICOMP] and [Parter-Yogev, ICALP '19], with recent progress on the matrix Spencer conjecture [Bansal-Meka-Jiang, STOC '23]. 2) We give an improved analysis of the constructions based on short-cycle decompositions, giving an m^{1+δ}-time algorithm for any constant δ > 0 for constructing Eulerian sparsifiers with O(nε^{-2}log³ n) edges.
Cite as

Sushant Sachdeva, Anvith Thudi, and Yibin Zhao. Better Sparsifiers for Directed Eulerian Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 119:1-119:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{sachdeva_et_al:LIPIcs.ICALP.2024.119,
  author =	{Sachdeva, Sushant and Thudi, Anvith and Zhao, Yibin},
  title =	{{Better Sparsifiers for Directed Eulerian Graphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{119:1--119:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.119},
  URN =		{urn:nbn:de:0030-drops-202628},
  doi =		{10.4230/LIPIcs.ICALP.2024.119},
  annote =	{Keywords: Graph algorithms, Linear algebra and computation, Discrepancy theory}
}
Document
DOI: 10.4230/DagMan.10.1.1
Current and Future Challenges in Knowledge Representation and Reasoning (Dagstuhl Perspectives Workshop 22282)

Authors: James P. Delgrande, Birte Glimm, Thomas Meyer, Miroslaw Truszczynski, and Frank Wolter

Published in: Dagstuhl Manifestos, Volume 10, Issue 1 (2024)

Abstract
Knowledge Representation and Reasoning is a central, longstanding, and active area of Artificial Intelligence. Over the years it has evolved significantly; more recently it has been challenged and complemented by research in areas such as machine learning and reasoning under uncertainty. In July 2022,sser a Dagstuhl Perspectives workshop was held on Knowledge Representation and Reasoning. The goal of the workshop was to describe the state of the art in the field, including its relation with other areas, its shortcomings and strengths, together with recommendations for future progress. We developed this manifesto based on the presentations, panels, working groups, and discussions that took place at the Dagstuhl Workshop. It is a declaration of our views on Knowledge Representation: its origins, goals, milestones, and current foci; its relation to other disciplines, especially to Artificial Intelligence; and on its challenges, along with key priorities for the next decade.
Cite as

James P. Delgrande, Birte Glimm, Thomas Meyer, Miroslaw Truszczynski, and Frank Wolter. Current and Future Challenges in Knowledge Representation and Reasoning (Dagstuhl Perspectives Workshop 22282). In Dagstuhl Manifestos, Volume 10, Issue 1, pp. 1-61, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{delgrande_et_al:DagMan.10.1.1,
  author =	{Delgrande, James P. and Glimm, Birte and Meyer, Thomas and Truszczynski, Miroslaw and Wolter, Frank},
  title =	{{Current and Future Challenges in Knowledge Representation and Reasoning (Dagstuhl Perspectives Workshop 22282)}},
  pages =	{1--61},
  journal =	{Dagstuhl Manifestos},
  ISSN =	{2193-2433},
  year =	{2024},
  volume =	{10},
  number =	{1},
  editor =	{Delgrande, James P. and Glimm, Birte and Meyer, Thomas and Truszczynski, Miroslaw and Wolter, Frank},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagMan.10.1.1},
  URN =		{urn:nbn:de:0030-drops-201403},
  doi =		{10.4230/DagMan.10.1.1},
  annote =	{Keywords: Knowledge representation and reasoning, Applications of logics, Declarative representations, Formal logic}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.43
Is the Algorithmic Kadison-Singer Problem Hard?

Authors: Ben Jourdan, Peter Macgregor, and He Sun

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
We study the following KS₂(c) problem: let c ∈ ℝ^+ be some constant, and v₁,…, v_m ∈ ℝ^d be vectors such that ‖v_i‖² ≤ α for any i ∈ [m] and ∑_{i=1}^m ⟨v_i, x⟩² = 1 for any x ∈ ℝ^d with ‖x‖ = 1. The KS₂(c) problem asks to find some S ⊂ [m], such that it holds for all x ∈ ℝ^d with ‖x‖ = 1 that |∑_{i∈S} ⟨v_i, x⟩² - 1/2| ≤ c⋅√α, or report no if such S doesn't exist. Based on the work of Marcus et al. [Adam Marcus et al., 2013] and Weaver [Nicholas Weaver, 2004], the KS₂(c) problem can be seen as the algorithmic Kadison-Singer problem with parameter c ∈ ℝ^+. Our first result is a randomised algorithm with one-sided error for the KS₂(c) problem such that (1) our algorithm finds a valid set S ⊂ [m] with probability at least 1-2/d, if such S exists, or (2) reports no with probability 1, if no valid sets exist. The algorithm has running time O(binom(m,n)⋅poly(m, d)) for n = O(d/ε² log(d) log(1/(c√α))), where ε is a parameter which controls the error of the algorithm. This presents the first algorithm for the Kadison-Singer problem whose running time is quasi-polynomial in m in a certain regime, although having exponential dependency on d. Moreover, it shows that the algorithmic Kadison-Singer problem is easier to solve in low dimensions. Our second result is on the computational complexity of the KS₂(c) problem. We show that the KS₂(1/(4√2)) problem is FNP-hard for general values of d, and solving the KS₂(1/(4√2)) problem is as hard as solving the NAE-3SAT problem.
Cite as

Ben Jourdan, Peter Macgregor, and He Sun. Is the Algorithmic Kadison-Singer Problem Hard?. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jourdan_et_al:LIPIcs.ISAAC.2023.43,
  author =	{Jourdan, Ben and Macgregor, Peter and Sun, He},
  title =	{{Is the Algorithmic Kadison-Singer Problem Hard?}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{43:1--43:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.43},
  URN =		{urn:nbn:de:0030-drops-193457},
  doi =		{10.4230/LIPIcs.ISAAC.2023.43},
  annote =	{Keywords: Kadison-Singer problem, spectral sparsification}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.103
The Support of Open Versus Closed Random Walks

Authors: Thomas Sauerwald, He Sun, and Danny Vagnozzi

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
A closed random walk of length 𝓁 on an undirected and connected graph G = (V,E) is a random walk that returns to the start vertex at step 𝓁, and its properties have been recently related to problems in different mathematical fields, e.g., geometry and combinatorics (Jiang et al., Annals of Mathematics '21) and spectral graph theory (McKenzie et al., STOC '21). For instance, in the context of analyzing the eigenvalue multiplicity of graph matrices, McKenzie et al. show that, with high probability, the support of a closed random walk of length 𝓁 ⩾ 1 is Ω(𝓁^{1/5}) on any bounded-degree graph, and leaves as an open problem whether a stronger bound of Ω(𝓁^{1/2}) holds for any regular graph. First, we show that the support of a closed random walk of length 𝓁 is at least Ω(𝓁^{1/2} / √{log n}) for any regular or bounded-degree graph on n vertices. Secondly, we prove for every 𝓁 ⩾ 1 the existence of a family of bounded-degree graphs, together with a start vertex such that the support is bounded by O(𝓁^{1/2}/√{log n}). Besides addressing the open problem of McKenzie et al., these two results also establish a subtle separation between closed random walks and open random walks, for which the support on any regular (or bounded-degree) graph is well-known to be Ω(𝓁^{1/2}) for all 𝓁 ⩾ 1. For irregular graphs, we prove that even if the start vertex is chosen uniformly, the support of a closed random walk may still be O(log 𝓁). This rules out a general polynomial lower bound in 𝓁 for all graphs. Finally, we apply our results on random walks to obtain new bounds on the multiplicity of the second largest eigenvalue of the adjacency matrices of graphs.
Cite as

Thomas Sauerwald, He Sun, and Danny Vagnozzi. The Support of Open Versus Closed Random Walks. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 103:1-103:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{sauerwald_et_al:LIPIcs.ICALP.2023.103,
  author =	{Sauerwald, Thomas and Sun, He and Vagnozzi, Danny},
  title =	{{The Support of Open Versus Closed Random Walks}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{103:1--103:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.103},
  URN =		{urn:nbn:de:0030-drops-181556},
  doi =		{10.4230/LIPIcs.ICALP.2023.103},
  annote =	{Keywords: support of random walks, eigenvalue multiplicity}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.25
Improved Bounds for Randomly Colouring Simple Hypergraphs

Authors: Weiming Feng, Heng Guo, and Jiaheng Wang

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

Abstract
We study the problem of sampling almost uniform proper q-colourings in k-uniform simple hypergraphs with maximum degree Δ. For any δ > 0, if k ≥ 20(1+δ)/δ and q ≥ 100Δ^({2+δ}/{k-4/δ-4}), the running time of our algorithm is Õ(poly(Δ k)⋅ n^1.01), where n is the number of vertices. Our result requires fewer colours than previous results for general hypergraphs (Jain, Pham, and Vuong, 2021; He, Sun, and Wu, 2021), and does not require Ω(log n) colours unlike the work of Frieze and Anastos (2017).
Cite as

Weiming Feng, Heng Guo, and Jiaheng Wang. Improved Bounds for Randomly Colouring Simple Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{feng_et_al:LIPIcs.APPROX/RANDOM.2022.25,
  author =	{Feng, Weiming and Guo, Heng and Wang, Jiaheng},
  title =	{{Improved Bounds for Randomly Colouring Simple Hypergraphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{25:1--25:17},
  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.2022.25},
  URN =		{urn:nbn:de:0030-drops-171477},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.25},
  annote =	{Keywords: Approximate counting, Markov chain, Mixing time, Hypergraph colouring}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.20
Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary

Authors: Aaron Bernstein, Jan van den Brand, Maximilian Probst Gutenberg, Danupon Nanongkai, Thatchaphol Saranurak, Aaron Sidford, and He Sun

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

Abstract
Designing efficient dynamic graph algorithms against an adaptive adversary is a major goal in the field of dynamic graph algorithms and has witnessed many exciting recent developments in, e.g., dynamic matching (Wajc STOC'20) and decremental shortest paths (Chuzhoy and Khanna STOC'19). Compared to other graph primitives (e.g. spanning trees and matchings), designing such algorithms for graph spanners and (more broadly) graph sparsifiers poses a unique challenge since there is no fast deterministic algorithm known for static computation and the lack of a way to adjust the output slowly (known as "small recourse/replacements"). This paper presents the first non-trivial efficient adaptive algorithms for maintaining many sparsifiers against an adaptive adversary. Specifically, we present algorithms that maintain 1) a polylog(n)-spanner of size Õ(n) in polylog(n) amortized update time, 2) an O(k)-approximate cut sparsifier of size Õ(n) in Õ(n^{1/k}) amortized update time, and 3) a polylog(n)-approximate spectral sparsifier in polylog(n) amortized update time. Our bounds are the first non-trivial ones even when only the recourse is concerned. Our results hold even against a stronger adversary, who can access the random bits previously used by the algorithms and the amortized update time of all algorithms can be made worst-case by paying sub-polynomial factors. Our spanner result resolves an open question by Ahmed et al. (2019) and our results and techniques imply additional improvements over existing results, including (i) answering open questions about decremental single-source shortest paths by Chuzhoy and Khanna (STOC'19) and Gutenberg and Wulff-Nilsen (SODA'20), implying a nearly-quadratic time algorithm for approximating minimum-cost unit-capacity flow and (ii) de-amortizing a result of Abraham et al. (FOCS'16) for dynamic spectral sparsifiers. Our results are based on two novel techniques. The first technique is a generic black-box reduction that allows us to assume that the graph is initially an expander with almost uniform-degree and, more importantly, stays as an almost uniform-degree expander while undergoing only edge deletions. The second technique is called proactive resampling: here we constantly re-sample parts of the input graph so that, independent of an adversary’s computational power, a desired structure of the underlying graph can be always maintained. Despite its simplicity, the analysis of this sampling scheme is far from trivial, because the adversary can potentially create dependencies between the random choices used by the algorithm. We believe these two techniques could be useful for developing other adaptive algorithms.
Cite as

Aaron Bernstein, Jan van den Brand, Maximilian Probst Gutenberg, Danupon Nanongkai, Thatchaphol Saranurak, Aaron Sidford, and He Sun. Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bernstein_et_al:LIPIcs.ICALP.2022.20,
  author =	{Bernstein, Aaron and van den Brand, Jan and Probst Gutenberg, Maximilian and Nanongkai, Danupon and Saranurak, Thatchaphol and Sidford, Aaron and Sun, He},
  title =	{{Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{20:1--20:20},
  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.2022.20},
  URN =		{urn:nbn:de:0030-drops-163611},
  doi =		{10.4230/LIPIcs.ICALP.2022.20},
  annote =	{Keywords: dynamic graph algorithm, adaptive adversary, spanner, sparsifier}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.25
Dynamic Inference in Probabilistic Graphical Models

Authors: Weiming Feng, Kun He, Xiaoming Sun, and Yitong Yin

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

Abstract
Probabilistic graphical models, such as Markov random fields (MRFs), are useful for describing high-dimensional distributions in terms of local dependence structures. The {probabilistic inference} is a fundamental problem related to graphical models, and sampling is a main approach for the problem. In this paper, we study probabilistic inference problems when the graphical model itself is changing dynamically with time. Such dynamic inference problems arise naturally in today’s application, e.g. multivariate time-series data analysis and practical learning procedures. We give a dynamic algorithm for sampling-based probabilistic inferences in MRFs, where each dynamic update can change the underlying graph and all parameters of the MRF simultaneously, as long as the total amount of changes is bounded. More precisely, suppose that the MRF has n variables and polylogarithmic-bounded maximum degree, and N(n) independent samples are sufficient for the inference for a polynomial function N(⋅). Our algorithm dynamically maintains an answer to the inference problem using Õ(n N(n)) space cost, and Õ(N(n) + n) incremental time cost upon each update to the MRF, as long as the Dobrushin-Shlosman condition is satisfied by the MRFs. This well-known condition has long been used for guaranteeing the efficiency of Markov chain Monte Carlo (MCMC) sampling in the traditional static setting. Compared to the static case, which requires Ω(n N(n)) time cost for redrawing all N(n) samples whenever the MRF changes, our dynamic algorithm gives a 𝛺^~(min{n, N(n)})-factor speedup. Our approach relies on a novel dynamic sampling technique, which transforms local Markov chains (a.k.a. single-site dynamics) to dynamic sampling algorithms, and an "algorithmic Lipschitz" condition that we establish for sampling from graphical models, namely, when the MRF changes by a small difference, samples can be modified to reflect the new distribution, with cost proportional to the difference on MRF.
Cite as

Weiming Feng, Kun He, Xiaoming Sun, and Yitong Yin. Dynamic Inference in Probabilistic Graphical Models. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{feng_et_al:LIPIcs.ITCS.2021.25,
  author =	{Feng, Weiming and He, Kun and Sun, Xiaoming and Yin, Yitong},
  title =	{{Dynamic Inference in Probabilistic Graphical Models}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{25:1--25:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.25},
  URN =		{urn:nbn:de:0030-drops-135643},
  doi =		{10.4230/LIPIcs.ITCS.2021.25},
  annote =	{Keywords: Dynamic inference, probabilistic graphical model, Gibbs sampling, Markov random filed}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.70
Augmenting the Algebraic Connectivity of Graphs

Authors: Bogdan-Adrian Manghiuc, Pan Peng, and He Sun

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
For any undirected graph G = (V,E) and a set E_W of candidate edges with E ∩ E_W = ∅, the (k,γ)-spectral augmentability problem is to find a set F of k edges from E_W with appropriate weighting, such that the algebraic connectivity of the resulting graph H = (V, E ∪ F) is least γ. Because of a tight connection between the algebraic connectivity and many other graph parameters, including the graph’s conductance and the mixing time of random walks in a graph, maximising the resulting graph’s algebraic connectivity by adding a small number of edges has been studied over the past 15 years, and has many practical applications in network optimisation. In this work we present an approximate and efficient algorithm for the (k,γ)-spectral augmentability problem, and our algorithm runs in almost-linear time under a wide regime of parameters. Our main algorithm is based on the following two novel techniques developed in the paper, which might have applications beyond the (k,γ)-spectral augmentability problem: - We present a fast algorithm for solving a feasibility version of an SDP for the algebraic connectivity maximisation problem from [Ghosh and Boyd, 2006]. Our algorithm is based on the classic primal-dual framework for solving SDP, which in turn uses the multiplicative weight update algorithm. We present a novel approach of unifying SDP constraints of different matrix and vector variables and give a good separation oracle accordingly. - We present an efficient algorithm for the subgraph sparsification problem, and for a wide range of parameters our algorithm runs in almost-linear time, in contrast to the previously best known algorithm running in at least Ω(n²mk) time [Kolla et al., 2010]. Our analysis shows how the randomised BSS framework can be generalised in the setting of subgraph sparsification, and how the potential functions can be applied to approximately keep track of different subspaces.
Cite as

Bogdan-Adrian Manghiuc, Pan Peng, and He Sun. Augmenting the Algebraic Connectivity of Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 70:1-70:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{manghiuc_et_al:LIPIcs.ESA.2020.70,
  author =	{Manghiuc, Bogdan-Adrian and Peng, Pan and Sun, He},
  title =	{{Augmenting the Algebraic Connectivity of Graphs}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{70:1--70:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.70},
  URN =		{urn:nbn:de:0030-drops-129367},
  doi =		{10.4230/LIPIcs.ESA.2020.70},
  annote =	{Keywords: Graph sparsification, Algebraic connectivity, Semidefinite programming}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.96
Contraction: A Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems

Authors: Shuai Shao and Yuxin Sun

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
We study complex zeros of the partition function of 2-spin systems, viewed as a multivariate polynomial in terms of the edge interaction parameters and the uniform external field. We obtain new zero-free regions in which all these parameters are complex-valued. Crucially based on the zero-freeness, we are able to extend the existence of correlation decay to these complex regions from real parameters. As a consequence, we obtain an FPTAS for computing the partition function of 2-spin systems on graphs of bounded degree for these parameter settings. We introduce the contraction property as a unified sufficient condition to devise FPTAS via either Weitz’s algorithm or Barvinok’s algorithm. Our main technical contribution is a very simple but general approach to extend any real parameter of which the 2-spin system exhibits correlation decay to its complex neighborhood where the partition function is zero-free and correlation decay still exists. This result formally establishes the inherent connection between two distinct notions of phase transition for 2-spin systems: the existence of correlation decay and the zero-freeness of the partition function via a unified perspective, contraction.
Cite as

Shuai Shao and Yuxin Sun. Contraction: A Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 96:1-96:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{shao_et_al:LIPIcs.ICALP.2020.96,
  author =	{Shao, Shuai and Sun, Yuxin},
  title =	{{Contraction: A Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{96:1--96:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.96},
  URN =		{urn:nbn:de:0030-drops-125036},
  doi =		{10.4230/LIPIcs.ICALP.2020.96},
  annote =	{Keywords: 2-Spin system, Correlation decay, Zero-freeness, Phase transition, Contraction}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.71
Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem

Authors: Huan Li, He Sun, and Luca Zanetti

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
We study spectral approaches for the MAX-2-LIN(k) problem, in which we are given a system of m linear equations of the form x_i - x_j is equivalent to c_{ij} mod k, and required to find an assignment to the n variables {x_i} that maximises the total number of satisfied equations. We consider Hermitian Laplacians related to this problem, and prove a Cheeger inequality that relates the smallest eigenvalue of a Hermitian Laplacian to the maximum number of satisfied equations of a MAX-2-LIN(k) instance I. We develop an O~(kn^2) time algorithm that, for any (1-epsilon)-satisfiable instance, produces an assignment satisfying a (1 - O(k)sqrt{epsilon})-fraction of equations. We also present a subquadratic-time algorithm that, when the graph associated with I is an expander, produces an assignment satisfying a (1- O(k^2)epsilon)-fraction of the equations. Our Cheeger inequality and first algorithm can be seen as generalisations of the Cheeger inequality and algorithm for MAX-CUT developed by Trevisan.
Cite as

Huan Li, He Sun, and Luca Zanetti. Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{li_et_al:LIPIcs.ESA.2019.71,
  author =	{Li, Huan and Sun, He and Zanetti, Luca},
  title =	{{Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{71:1--71:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola 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.2019.71},
  URN =		{urn:nbn:de:0030-drops-111926},
  doi =		{10.4230/LIPIcs.ESA.2019.71},
  annote =	{Keywords: Spectral methods, Hermitian Laplacians, the Max-2-Lin problem, Unique Games}
}
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