Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)
Praneeth Kacham and David P. Woodruff. Faster Algorithms for Schatten-p Low Rank Approximation. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{kacham_et_al:LIPIcs.APPROX/RANDOM.2024.55, author = {Kacham, Praneeth and Woodruff, David P.}, title = {{Faster Algorithms for Schatten-p Low Rank Approximation}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {55:1--55:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.55}, URN = {urn:nbn:de:0030-drops-210488}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.55}, annote = {Keywords: Low Rank Approximation, Schatten Norm, Rectangular Matrix Multiplication, Stability Analysis} }
Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)
Yotam Dikstein and Irit Dinur. Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 62:1-62:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{dikstein_et_al:LIPIcs.APPROX/RANDOM.2024.62, author = {Dikstein, Yotam and Dinur, Irit}, title = {{Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {62:1--62:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.62}, URN = {urn:nbn:de:0030-drops-210556}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.62}, annote = {Keywords: High Dimensional Expanders, HDX, Spectral Expansion, Coboundary Expansion, Cocycle Expansion, Cosystolic Expansion} }
Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)
Inbar Ben Yaacov, Yotam Dikstein, and Gal Maor. Sparse High Dimensional Expanders via Local Lifts. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 68:1-68:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{benyaacov_et_al:LIPIcs.APPROX/RANDOM.2024.68, author = {Ben Yaacov, Inbar and Dikstein, Yotam and Maor, Gal}, title = {{Sparse High Dimensional Expanders via Local Lifts}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {68:1--68:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.68}, URN = {urn:nbn:de:0030-drops-210612}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.68}, annote = {Keywords: High Dimensional Expanders, HDX, Spectral Expansion, Lifts, Covers, Explicit Constructions, Randomized Constructions, Deterministic Constructions} }
Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)
Noga Alon, Allan Grønlund, Søren Fuglede Jørgensen, and Kasper Green Larsen. Sublinear Time Shortest Path in Expander Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{alon_et_al:LIPIcs.MFCS.2024.8, author = {Alon, Noga and Gr{\o}nlund, Allan and J{\o}rgensen, S{\o}ren Fuglede and Larsen, Kasper Green}, title = {{Sublinear Time Shortest Path in Expander Graphs}}, booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)}, pages = {8:1--8:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-335-5}, ISSN = {1868-8969}, year = {2024}, volume = {306}, editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.8}, URN = {urn:nbn:de:0030-drops-205646}, doi = {10.4230/LIPIcs.MFCS.2024.8}, annote = {Keywords: Shortest Path, Expanders, Breadth First Search, Graph Algorithms} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Cella Florescu, Rasmus Kyng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Optimal Electrical Oblivious Routing on Expanders. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 65:1-65:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{florescu_et_al:LIPIcs.ICALP.2024.65, author = {Florescu, Cella and Kyng, Rasmus and Gutenberg, Maximilian Probst and Sachdeva, Sushant}, title = {{Optimal Electrical Oblivious Routing on Expanders}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {65:1--65:19}, 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.2024.65}, URN = {urn:nbn:de:0030-drops-202083}, doi = {10.4230/LIPIcs.ICALP.2024.65}, annote = {Keywords: Expanders, Oblivious routing for 𝓁\underlinep, Electrical flow routing} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Yotam Kenneth and Robert Krauthgamer. Cut Sparsification and Succinct Representation of Submodular Hypergraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 97:1-97:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{kenneth_et_al:LIPIcs.ICALP.2024.97, author = {Kenneth, Yotam and Krauthgamer, Robert}, title = {{Cut Sparsification and Succinct Representation of Submodular Hypergraphs}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {97:1--97:17}, 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.2024.97}, URN = {urn:nbn:de:0030-drops-202406}, doi = {10.4230/LIPIcs.ICALP.2024.97}, annote = {Keywords: Cut Sparsification, Submodular Hypergraphs, Succinct Representation} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Sanjeev Khanna, Aaron (Louie) Putterman, and Madhu Sudan. Almost-Tight Bounds on Preserving Cuts in Classes of Submodular Hypergraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 98:1-98:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{khanna_et_al:LIPIcs.ICALP.2024.98, author = {Khanna, Sanjeev and Putterman, Aaron (Louie) and Sudan, Madhu}, title = {{Almost-Tight Bounds on Preserving Cuts in Classes of Submodular Hypergraphs}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {98:1--98:17}, 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.2024.98}, URN = {urn:nbn:de:0030-drops-202410}, doi = {10.4230/LIPIcs.ICALP.2024.98}, annote = {Keywords: Sparsification, sketching, hypergraphs} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Martin Grohe and Daniel Neuen. Isomorphism for Tournaments of Small Twin Width. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 78:1-78:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{grohe_et_al:LIPIcs.ICALP.2024.78, author = {Grohe, Martin and Neuen, Daniel}, title = {{Isomorphism for Tournaments of Small Twin Width}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {78:1--78:20}, 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.2024.78}, URN = {urn:nbn:de:0030-drops-202216}, doi = {10.4230/LIPIcs.ICALP.2024.78}, annote = {Keywords: tournament isomorphism, twin width, fixed-parameter tractability, Weisfeiler-Leman algorithm} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Leonid Gurvits, Nathan Klein, and Jonathan Leake. From Trees to Polynomials and Back Again: New Capacity Bounds with Applications to TSP. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 79:1-79:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{gurvits_et_al:LIPIcs.ICALP.2024.79, author = {Gurvits, Leonid and Klein, Nathan and Leake, Jonathan}, title = {{From Trees to Polynomials and Back Again: New Capacity Bounds with Applications to TSP}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {79:1--79:20}, 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.2024.79}, URN = {urn:nbn:de:0030-drops-202229}, doi = {10.4230/LIPIcs.ICALP.2024.79}, annote = {Keywords: traveling salesman problem, strongly Rayleigh distributions, polynomial capacity, probability lower bounds, combinatorial lower bounds} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Orestis Plevrakis, Seyoon Ragavan, and S. Matthew Weinberg. On the Cut-Query Complexity of Approximating Max-Cut. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 115:1-115:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{plevrakis_et_al:LIPIcs.ICALP.2024.115, author = {Plevrakis, Orestis and Ragavan, Seyoon and Weinberg, S. Matthew}, title = {{On the Cut-Query Complexity of Approximating Max-Cut}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {115:1--115:20}, 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.2024.115}, URN = {urn:nbn:de:0030-drops-202587}, doi = {10.4230/LIPIcs.ICALP.2024.115}, annote = {Keywords: query complexity, maximum cut, approximation algorithms, graph sparsification} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
Kent Quanrud. Adaptive Sparsification for Matroid Intersection. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 118:1-118:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
@InProceedings{quanrud:LIPIcs.ICALP.2024.118, author = {Quanrud, Kent}, title = {{Adaptive Sparsification for Matroid Intersection}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {118:1--118:20}, 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.2024.118}, URN = {urn:nbn:de:0030-drops-202614}, doi = {10.4230/LIPIcs.ICALP.2024.118}, annote = {Keywords: Matroid intersection, adaptive sparsification, multiplicative-weight udpates, primal-dual} }
Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)
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)
@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} }
Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)
Papri Dey, Ravi Kannan, Nick Ryder, and Nikhil Srivastava. Bit Complexity of Jordan Normal Form and Polynomial Spectral Factorization. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
@InProceedings{dey_et_al:LIPIcs.ITCS.2023.42, author = {Dey, Papri and Kannan, Ravi and Ryder, Nick and Srivastava, Nikhil}, title = {{Bit Complexity of Jordan Normal Form and Polynomial Spectral Factorization}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {42:1--42:18}, 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.2023.42}, URN = {urn:nbn:de:0030-drops-175450}, doi = {10.4230/LIPIcs.ITCS.2023.42}, annote = {Keywords: Symbolic algorithms, numerical algorithms, linear algebra} }
Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Hariharan Narayanan, Rikhav Shah, and Nikhil Srivastava. A Spectral Approach to Polytope Diameter. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 108:1-108:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
@InProceedings{narayanan_et_al:LIPIcs.ITCS.2022.108, author = {Narayanan, Hariharan and Shah, Rikhav and Srivastava, Nikhil}, title = {{A Spectral Approach to Polytope Diameter}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {108:1--108:22}, 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.2022.108}, URN = {urn:nbn:de:0030-drops-157044}, doi = {10.4230/LIPIcs.ITCS.2022.108}, annote = {Keywords: Polytope diameter, Markov Chain} }
Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)
Theo McKenzie and Sidhanth Mohanty. High-Girth Near-Ramanujan Graphs with Lossy Vertex Expansion. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 96:1-96:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
@InProceedings{mckenzie_et_al:LIPIcs.ICALP.2021.96, author = {McKenzie, Theo and Mohanty, Sidhanth}, title = {{High-Girth Near-Ramanujan Graphs with Lossy Vertex Expansion}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {96:1--96:15}, 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.2021.96}, URN = {urn:nbn:de:0030-drops-141655}, doi = {10.4230/LIPIcs.ICALP.2021.96}, annote = {Keywords: expander graphs, Ramanujan graphs, vertex expansion, girth} }
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