6 Search Results for "Cai, Chen"

Document
DOI: 10.4230/LIPIcs.ESA.2023.29
Effective Resistances in Non-Expander Graphs

Authors: Dongrun Cai, Xue Chen, and Pan Peng

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

Abstract
Effective resistances are ubiquitous in graph algorithms and network analysis. For an undirected graph G, its effective resistance R_G(s,t) between two vertices s and t is defined as the equivalent resistance between s and t if G is thought of as an electrical network with unit resistance on each edge. If we use L_G to denote the Laplacian matrix of G and L_G^† to denote its pseudo-inverse, we have R_G(s,t) = (𝟏_s-𝟏_t)^⊤ L^† (𝟏_s-𝟏_t) such that classical Laplacian solvers [Daniel A. Spielman and Shang{-}Hua Teng, 2014] provide almost-linear time algorithms to approximate R_G(s,t). In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair s and t. We consider the classical adjacency list model [Ron, 2019] for local algorithms. While recent works [Andoni et al., 2018; Peng et al., 2021; Li and Sachdeva, 2023] have provided sublinear time algorithms for expander graphs, we prove several lower bounds for general graphs of n vertices and m edges: 1) It needs Ω(n) queries to obtain 1.01-approximations of the effective resistance of an adjacent pair s and t, even for graphs of degree at most 3 except s and t. 2) For graphs of degree at most d and any parameter 𝓁, it needs Ω(m/𝓁) queries to obtain c ⋅ min{d,𝓁}-approximations where c > 0 is a universal constant. Moreover, we supplement the first lower bound by providing a sublinear time (1+ε)-approximation algorithm for graphs of degree 2 except the pair s and t. One of our technical ingredients is to bound the expansion of a graph in terms of the smallest non-trivial eigenvalue of its Laplacian matrix after removing edges. We discover a new lower bound on the eigenvalues of perturbed graphs (resp. perturbed matrices) by incorporating the effective resistance of the removed edge (resp. the leverage scores of the removed rows), which may be of independent interest.
Cite as

Dongrun Cai, Xue Chen, and Pan Peng. Effective Resistances in Non-Expander Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cai_et_al:LIPIcs.ESA.2023.29,
  author =	{Cai, Dongrun and Chen, Xue and Peng, Pan},
  title =	{{Effective Resistances in Non-Expander Graphs}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{29:1--29:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.29},
  URN =		{urn:nbn:de:0030-drops-186823},
  doi =		{10.4230/LIPIcs.ESA.2023.29},
  annote =	{Keywords: Sublinear Time Algorithm, Effective Resistance, Leverage Scores, Matrix Perturbation}
}
Document
DOI: 10.4230/LIPIcs.SEA.2021.14
Approximation Algorithms for 1-Wasserstein Distance Between Persistence Diagrams

Authors: Samantha Chen and Yusu Wang

Published in: LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

Abstract
Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially complex input object (be it a graph, an image, or a point set and so on) to a unified type of feature summary, called the persistence diagrams. One can then carry out downstream data analysis tasks using such persistence diagram representations. A key problem is to compute the distance between two persistence diagrams efficiently. In particular, a persistence diagram is essentially a multiset of points in the plane, and one popular distance is the so-called 1-Wasserstein distance between persistence diagrams. In this paper, we present two algorithms to approximate the 1-Wasserstein distance for persistence diagrams in near-linear time. These algorithms primarily follow the same ideas as two existing algorithms to approximate optimal transport between two finite point-sets in Euclidean spaces via randomly shifted quadtrees. We show how these algorithms can be effectively adapted for the case of persistence diagrams. Our algorithms are much more efficient than previous exact and approximate algorithms, both in theory and in practice, and we demonstrate its efficiency via extensive experiments. They are conceptually simple and easy to implement, and the code is publicly available in github.
Cite as

Samantha Chen and Yusu Wang. Approximation Algorithms for 1-Wasserstein Distance Between Persistence Diagrams. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chen_et_al:LIPIcs.SEA.2021.14,
  author =	{Chen, Samantha and Wang, Yusu},
  title =	{{Approximation Algorithms for 1-Wasserstein Distance Between Persistence Diagrams}},
  booktitle =	{19th International Symposium on Experimental Algorithms (SEA 2021)},
  pages =	{14:1--14:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-185-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{190},
  editor =	{Coudert, David and Natale, Emanuele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.14},
  URN =		{urn:nbn:de:0030-drops-137861},
  doi =		{10.4230/LIPIcs.SEA.2021.14},
  annote =	{Keywords: persistence diagrams, approximation algorithms, Wasserstein distance, optimal transport}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.26
Elder-Rule-Staircodes for Augmented Metric Spaces

Authors: Chen Cai, Woojin Kim, Facundo Mémoli, and Yusu Wang

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
An augmented metric space (X, d_X, f_X) is a metric space (X, d_X) equipped with a function f_X: X → ℝ. It arises commonly in practice, e.g, a point cloud X in ℝ^d where each point x∈ X has a density function value f_X(x) associated to it. Such an augmented metric space naturally gives rise to a 2-parameter filtration. However, the resulting 2-parameter persistence module could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of a 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode the zeroth homology of the 2-parameter filtration induced by a finite augmented metric space. Specifically, given a finite (X, d_X, f_X), its elder-rule-staircode consists of n = |X| number of staircase-like blocks in the plane. We show that the fibered barcode, the fibered merge tree, and the graded Betti numbers associated to the zeroth homology of the 2-parameter filtration induced by (X, d_X, f_X) can all be efficiently computed once the elder-rule-staircode is given. Furthermore, for certain special cases, this staircode corresponds exactly to the set of indecomposables of the zeroth homology of the 2-parameter filtration. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O(n²log n) time, which can be improved to O(n²α(n)) if X is from a fixed dimensional Euclidean space ℝ^d, where α(n) is the inverse Ackermann function.
Cite as

Chen Cai, Woojin Kim, Facundo Mémoli, and Yusu Wang. Elder-Rule-Staircodes for Augmented Metric Spaces. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cai_et_al:LIPIcs.SoCG.2020.26,
  author =	{Cai, Chen and Kim, Woojin and M\'{e}moli, Facundo and Wang, Yusu},
  title =	{{Elder-Rule-Staircodes for Augmented Metric Spaces}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.26},
  URN =		{urn:nbn:de:0030-drops-121848},
  doi =		{10.4230/LIPIcs.SoCG.2020.26},
  annote =	{Keywords: Persistent homology, Multiparameter persistence, Barcodes, Elder rule, Hierarchical clustering, Graded Betti numbers}
}
Document
DOI: 10.4230/LIPIcs.CCC.2017.19
A Quadratic Lower Bound for Homogeneous Algebraic Branching Programs

Authors: Mrinal Kumar

Published in: LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

Abstract
An algebraic branching program (ABP) is a directed acyclic graph, with a start vertex s, and end vertex t and each edge having a weight which is an affine form in variables x_1, x_2, ..., x_n over an underlying field. An ABP computes a polynomial in a natural way, as the sum of weights of all paths from s to t, where the weight of a path is the product of the weights of the edges in the path. An ABP is said to be homogeneous if the polynomial computed at every vertex is homogeneous. In this paper, we show that any homogeneous algebraic branching program which computes the polynomial x_1^n + x_2^n + ... + x_n^n has at least Omega(n^2) vertices (and edges). To the best of our knowledge, this seems to be the first non-trivial super-linear lower bound on the number of vertices for a general homogeneous ABP and slightly improves the known lower bound of Omega(n log n) on the number of edges in a general (possibly non-homogeneous) ABP, which follows from the classical results of Strassen (1973) and Baur--Strassen (1983). On the way, we also get an alternate and unified proof of an Omega(n log n) lower bound on the size of a homogeneous arithmetic circuit (follows from [Strassen, 1973] and [Baur-Strassen, 1983]), and an n/2 lower bound (n over reals) on the determinantal complexity of an explicit polynomial [Mignon-Ressayre, 2004], [Cai, Chen, Li, 2010], [Yabe, 2015]. These are currently the best lower bounds known for these problems for any explicit polynomial, and were originally proved nearly two decades apart using seemingly different proof techniques.
Cite as

Mrinal Kumar. A Quadratic Lower Bound for Homogeneous Algebraic Branching Programs. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kumar:LIPIcs.CCC.2017.19,
  author =	{Kumar, Mrinal},
  title =	{{A Quadratic Lower Bound for Homogeneous Algebraic Branching Programs}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{O'Donnell, Ryan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.19},
  URN =		{urn:nbn:de:0030-drops-75134},
  doi =		{10.4230/LIPIcs.CCC.2017.19},
  annote =	{Keywords: algebraic branching programs, arithmetic circuits, determinantal complexity, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.STACS.2014.239
Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach

Authors: Suryajith Chillara and Partha Mukhopadhyay

Published in: LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

Abstract
Tavenas has recently proved that any n^{O(1)}-variate and degree n polynomial in VP can be computed by a depth-4 SigmaPi^[O(sqrt{n})]SigmaPi^{[sqrt{n}]} circuit of size 2^{O(n^{1/2}.log(n))} [Tavenas, 2013]. So, to prove that VP is not equal to VNP it is sufficient to show that an explicit polynomial in VNP of degree n requires 2^{omega(n^{1/2}.log(n))} size depth-4 circuits. Soon after Tavenas' result, for two different explicit polynomials, depth-4 circuit size lower bounds of 2^{Omega(n^{1/2}.log(n))} have been proved (see [Kayal, Saha, and Saptharishi, 2013] and [Fournier et al., 2013]). In particular, using combinatorial design [Kayal et al., 2013] construct an explicit polynomial in VNP that requires depth-4 circuits of size 2^{Omega(n^{1/2}.log(n))} and [Fournier et al., 2013] show that the iterated matrix multiplication polynomial (which is in VP) also requires 2^{Omega(n^{1/2}.log(n))} size depth-4 circuits. In this paper, we identify a simple combinatorial property such that any polynomial f that satisfies this property would achieve a similar depth-4 circuit size lower bound. In particular, it does not matter whether f is in VP or in VNP. As a result, we get a simple unified lower bound analysis for the above mentioned polynomials. Another goal of this paper is to compare our current knowledge of the depth-4 circuit size lower bounds and the determinantal complexity lower bounds. Currently the best known determinantal complexity lower bound is Omega(n^2) for Permanent of a nxn matrix (which is a n^2-variate and degree n polynomial) [Cai, Chen, and Li, 2008]. We prove that the determinantal complexity of the iterated matrix multiplication polynomial is Omega(dn) where d is the number of matrices and n is the dimension of the matrices. So for d=n, we get that the iterated matrix multiplication polynomial achieves the current best known lower bounds in both fronts: depth-4 circuit size and determinantal complexity. Our result also settles the determinantal complexity of the iterated matrix multiplication polynomial to Theta(dn). To the best of our knowledge, a Theta(n) bound for the determinantal complexity for the iterated matrix multiplication polynomial was known only for any constant d>1 [Jansen, 2011].
Cite as

Suryajith Chillara and Partha Mukhopadhyay. Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 239-250, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{chillara_et_al:LIPIcs.STACS.2014.239,
  author =	{Chillara, Suryajith and Mukhopadhyay, Partha},
  title =	{{Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{239--250},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Mayr, Ernst W. and Portier, Natacha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.239},
  URN =		{urn:nbn:de:0030-drops-44610},
  doi =		{10.4230/LIPIcs.STACS.2014.239},
  annote =	{Keywords: Arithmetic Circuits, Determinantal Complexity, Depth-4 Lower Bounds}
}
Document
DOI: 10.4230/LIPIcs.STACS.2009.1851
Polynomial Kernelizations for MIN F^+Pi_1 and MAX NP

Authors: Stefan Kratsch

Published in: LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Abstract
The relation of constant-factor approximability to fixed-parameter tractability and kernelization is a long-standing open question. We prove that two large classes of constant-factor approximable problems, namely~$\textsc{MIN F}^+\Pi_1$ and~$\textsc{MAX NP}$, including the well-known subclass~$\textsc{MAX SNP}$, admit polynomial kernelizations for their natural decision versions. This extends results of Cai and Chen (JCSS 1997), stating that the standard parameterizations of problems in~$\textsc{MAX SNP}$ and~$\textsc{MIN F}^+\Pi_1$ are fixed-parameter tractable, and complements recent research on problems that do not admit polynomial kernelizations (Bodlaender et al.\ ICALP 2008).
Cite as

Stefan Kratsch. Polynomial Kernelizations for MIN F^+Pi_1 and MAX NP. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 601-612, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{kratsch:LIPIcs.STACS.2009.1851,
  author =	{Kratsch, Stefan},
  title =	{{Polynomial Kernelizations for MIN F^+Pi\underline1 and MAX NP}},
  booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{601--612},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-09-5},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{3},
  editor =	{Albers, Susanne and Marion, Jean-Yves},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1851},
  URN =		{urn:nbn:de:0030-drops-18511},
  doi =		{10.4230/LIPIcs.STACS.2009.1851},
  annote =	{Keywords: Parameterized complexity, Kernelization, Approximation algorithms}
}
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