15 Search Results for "Hopkins, Max"

Document
DOI: 10.4230/LIPIcs.ITP.2025.9
Algebra Is Half the Battle: Verifying Presentations of Graded Unipotent Chevalley Groups

Authors: Eric Wang, Arohee Bhoja, Cayden Codel, and Noah G. Singer

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
Graded unipotent Chevalley groups are an important family of groups on matrices with polynomial entries over a finite field. Using the Lean theorem prover, we verify that three such groups, namely, the A₃- and the two B₃-type groups, satisfy a useful group-theoretic condition. Specifically, these groups are defined by a set of equations called Steinberg relations, and we prove that a certain canonical "smaller" set of Steinberg relations suffices to derive the rest. Our work is motivated by an application for building topologically-interesting objects called higher-dimensional expanders (HDXs). In the past decade, HDXs have formed the basis for many new results in theoretical computer science, such as in quantum error correction and in property testing. Yet despite the increasing prevalence of HDXs, only two methods of constructing them are known. One such method builds an HDX from groups that satisfy the aforementioned property, and the Chevalley groups we use are (essentially) the only ones currently known to satisfy it.
Cite as

Eric Wang, Arohee Bhoja, Cayden Codel, and Noah G. Singer. Algebra Is Half the Battle: Verifying Presentations of Graded Unipotent Chevalley Groups. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{wang_et_al:LIPIcs.ITP.2025.9,
  author =	{Wang, Eric and Bhoja, Arohee and Codel, Cayden and Singer, Noah G.},
  title =	{{Algebra Is Half the Battle: Verifying Presentations of Graded Unipotent Chevalley Groups}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.9},
  URN =		{urn:nbn:de:0030-drops-246071},
  doi =		{10.4230/LIPIcs.ITP.2025.9},
  annote =	{Keywords: Group presentations, term rewriting, metaprogramming, proof automation, the Lean theorem prover}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.11
On Finding Randomly Planted Cliques in Arbitrary Graphs

Authors: Francesco Agrimonti, Marco Bressan, and Tommaso d'Orsi

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We study a planted clique model introduced by Feige [Uriel Feige, 2021] where a complete graph of size c⋅ n is planted uniformly at random in an arbitrary n-vertex graph. We give a simple deterministic algorithm that, in almost linear time, recovers a clique of size (c/3)^O(1/c) ⋅ n as long as the original graph has maximum degree at most (1-p)n for some fixed p > 0. The proof hinges on showing that the degrees of the final graph are correlated with the planted clique, in a way similar to (but more intricate than) the classical G(n,1/2)+K_√n planted clique model. Our algorithm suggests a separation from the worst-case model, where, assuming the Unique Games Conjecture, no polynomial algorithm can find cliques of size Ω(n) for every fixed c > 0, even if the input graph has maximum degree (1-p)n. Our techniques extend beyond the planted clique model. For example, when the planted graph is a balanced biclique, we recover a balanced biclique of size larger than the best guarantees known for the worst case.
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Francesco Agrimonti, Marco Bressan, and Tommaso d'Orsi. On Finding Randomly Planted Cliques in Arbitrary Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{agrimonti_et_al:LIPIcs.APPROX/RANDOM.2025.11,
  author =	{Agrimonti, Francesco and Bressan, Marco and d'Orsi, Tommaso},
  title =	{{On Finding Randomly Planted Cliques in Arbitrary Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{11:1--11:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.11},
  URN =		{urn:nbn:de:0030-drops-243774},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.11},
  annote =	{Keywords: Computational Complexity, Planted Clique, Semi-random, Unique Games Conjecture, Approximation Algorithms}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.16
Sparsest Cut and Eigenvalue Multiplicities on Low Degree Abelian Cayley Graphs

Authors: Tommaso d'Orsi, Chris Jones, Jake Ruotolo, Salil Vadhan, and Jiyu Zhang

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
Whether or not the Sparsest Cut problem admits an efficient O(1)-approximation algorithm is a fundamental algorithmic question with connections to geometry and the Unique Games Conjecture. Revisiting spectral algorithms for Sparsest Cut, we present a novel, simple algorithm that combines eigenspace enumeration with a new algorithm for the Cut Improvement problem. The runtime of our algorithm is parametrized by a quantity that we call the solution dimension SD_ε(G): the smallest k such that the subspace spanned by the first k Laplacian eigenvectors contains all but ε fraction of a sparsest cut. Our algorithm matches the guarantees of prior methods based on the threshold-rank paradigm, while also extending beyond them. To illustrate this, we study its performance on low degree Cayley graphs over Abelian groups - canonical examples of graphs with poor expansion properties. We prove that low degree Abelian Cayley graphs have small solution dimension, yielding an algorithm that computes a (1+ε)-approximation to the uniform Sparsest Cut of a degree-d Cayley graph over an Abelian group of size n in time n^O(1) ⋅ exp{(d/ε)^O(d)}. Along the way to bounding the solution dimension of Abelian Cayley graphs, we analyze their sparse cuts and spectra, proving that the collection of O(1)-approximate sparsest cuts has an ε-net of size exp{(d/ε)^O(d)} and that the multiplicity of λ₂ is bounded by 2^O(d). The latter bound is tight and improves on a previous bound of 2^O(d²) by Lee and Makarychev.
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Tommaso d'Orsi, Chris Jones, Jake Ruotolo, Salil Vadhan, and Jiyu Zhang. Sparsest Cut and Eigenvalue Multiplicities on Low Degree Abelian Cayley Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 16:1-16:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dorsi_et_al:LIPIcs.APPROX/RANDOM.2025.16,
  author =	{d'Orsi, Tommaso and Jones, Chris and Ruotolo, Jake and Vadhan, Salil and Zhang, Jiyu},
  title =	{{Sparsest Cut and Eigenvalue Multiplicities on Low Degree Abelian Cayley Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{16:1--16:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.16},
  URN =		{urn:nbn:de:0030-drops-243827},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.16},
  annote =	{Keywords: Sparsest Cut, Spectral Graph Theory, Cayley Graphs, Approximation Algorithms}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.25
Quantum LDPC Codes of Almost Linear Distance via Iterated Homological Products

Authors: Louis Golowich and Venkatesan Guruswami

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
The first linear-distance quantum LDPC codes were recently constructed by a line of breakthrough works (culminating in the result of Panteleev & Kalachev, 2021). All such constructions, even when allowing for almost-linear distance, are based on an operation called a balanced (or lifted) product, which is used in a one-shot manner to combine a pair of large classical codes possessing a group symmetry. We present a new construction of almost-linear distance quantum LDPC codes that is iterative in nature. Our construction is based on a more basic and widely used product, namely the homological product (i.e. the tensor product of chain complexes). Specifically, for every ε > 0, we obtain a family of [[N,N^{1-ε},N^{1-ε}]] (subsystem) quantum LDPC codes via repeated homological products of a constant-sized quantum locally testable code. Our key idea is to remove certain low-weight codewords using subsystem codes (while still maintaining constant stabilizer weight), in order to circumvent a particular obstruction that limited the distance of many prior homological product code constructions to at most Õ(√N).
Cite as

Louis Golowich and Venkatesan Guruswami. Quantum LDPC Codes of Almost Linear Distance via Iterated Homological Products. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 25:1-25:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{golowich_et_al:LIPIcs.CCC.2025.25,
  author =	{Golowich, Louis and Guruswami, Venkatesan},
  title =	{{Quantum LDPC Codes of Almost Linear Distance via Iterated Homological Products}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{25:1--25:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.25},
  URN =		{urn:nbn:de:0030-drops-237196},
  doi =		{10.4230/LIPIcs.CCC.2025.25},
  annote =	{Keywords: Quantum Error Correction, Quantum LDPC Code, Homological Product, Iterative Construction}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.7
Sparser Abelian High Dimensional Expanders

Authors: Yotam Dikstein, Siqi Liu, and Avi Wigderson

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
The focus of this paper is the development of new elementary techniques for the construction and analysis of high dimensional expanders. Specifically, we present two new explicit constructions of Cayley high dimensional expanders (HDXs) over the abelian group 𝔽₂ⁿ. Our expansion proofs use only linear algebra and combinatorial arguments. The first construction gives local spectral HDXs of any constant dimension and subpolynomial degree exp(n^ε) for every ε > 0, improving on a construction by Golowich [Golowich, 2023] which achieves ε = 1/2. [Golowich, 2023] derives these HDXs by sparsifying the complete Grassmann poset of subspaces. The novelty in our construction is the ability to sparsify any expanding Grassmann posets, leading to iterated sparsification and much smaller degrees. The sparse Grassmannian (which is of independent interest in the theory of HDXs) serves as the generating set of the Cayley graph. Our second construction gives a 2-dimensional HDX of any polynomial degree exp(ε n) for any constant ε > 0, which is simultaneously a spectral expander and a coboundary expander. To the best of our knowledge, this is the first such non-trivial construction. We name it the Johnson complex, as it is derived from the classical Johnson scheme, whose vertices serve as the generating set of this Cayley graph. This construction may be viewed as a derandomization of the recent random geometric complexes of [Liu et al., 2023]. Establishing coboundary expansion through Gromov’s "cone method" and the associated isoperimetric inequalities is the most intricate aspect of this construction. While these two constructions are quite different, we show that they both share a common structure, resembling the intersection patterns of vectors in the Hadamard code. We propose a general framework of such "Hadamard-like" constructions in the hope that it will yield new HDXs.
Cite as

Yotam Dikstein, Siqi Liu, and Avi Wigderson. Sparser Abelian High Dimensional Expanders. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 7:1-7:98, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dikstein_et_al:LIPIcs.CCC.2025.7,
  author =	{Dikstein, Yotam and Liu, Siqi and Wigderson, Avi},
  title =	{{Sparser Abelian High Dimensional Expanders}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{7:1--7:98},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.7},
  URN =		{urn:nbn:de:0030-drops-237013},
  doi =		{10.4230/LIPIcs.CCC.2025.7},
  annote =	{Keywords: Local spectral expander, coboundary expander, Grassmannian expander}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.27
The Randomness Complexity of Differential Privacy

Authors: Clément L. Canonne, Francis E. Su, and Salil P. Vadhan

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We initiate the study of the randomness complexity of differential privacy, i.e., how many random bits an algorithm needs in order to generate accurate differentially private releases. As a test case, we focus on the task of releasing the results of d counting queries, or equivalently all one-way marginals on a d-dimensional dataset with boolean attributes. While standard differentially private mechanisms for this task have randomness complexity that grows linearly with d, we show that, surprisingly, only log₂ d+O(1) random bits (in expectation) suffice to achieve an error that depends polynomially on d (and is independent of the size n of the dataset), and furthermore this is possible with pure, unbounded differential privacy and privacy-loss parameter ε = 1/poly(d). Conversely, we show that at least log₂ d-O(1) random bits are also necessary for nontrivial accuracy, even with approximate, bounded DP, provided the privacy-loss parameters satisfy ε,δ ≤ 1/poly(d). We obtain our results by establishing a close connection between the randomness complexity of differentially private mechanisms and the geometric notion of "deterministic rounding schemes" recently introduced and studied by Vander Woude et al. (2022, 2023).
Cite as

Clément L. Canonne, Francis E. Su, and Salil P. Vadhan. The Randomness Complexity of Differential Privacy. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 27:1-27:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{canonne_et_al:LIPIcs.ITCS.2025.27,
  author =	{Canonne, Cl\'{e}ment L. and Su, Francis E. and Vadhan, Salil P.},
  title =	{{The Randomness Complexity of Differential Privacy}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{27:1--27:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.27},
  URN =		{urn:nbn:de:0030-drops-226556},
  doi =		{10.4230/LIPIcs.ITCS.2025.27},
  annote =	{Keywords: differential privacy, randomness, geometry}
}
Document
DOI: 10.4230/LIPIcs.AofA.2024.28
Sharpened Localization of the Trailing Point of the Pareto Record Frontier

Authors: James Allen Fill, Daniel Q. Naiman, and Ao Sun

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
For d ≥ 2 and i.i.d. d-dimensional observations X^{(1)}, X^{(2)}, … with independent Exponential(1) coordinates, we revisit the study by Fill and Naiman (Electron. J. Probab., 25:Paper No. 92, 24 pp., 2020) of the boundary (relative to the closed positive orthant), or "frontier", F_n of the closed Pareto record-setting (RS) region RS_n := {0 ≤ x ∈ R^d: x ⊀ X^(i) for all 1 ≤ i ≤ n} at time n, where 0 ≤ x means that 0 ≤ x_j for 1 ≤ j ≤ d and x ≺ y means that x_j < y_j for 1 ≤ j ≤ d. With x_+ : = ∑_{j = 1}^d x_j = ‖x‖₁, let F_n^- := min{x_+: x ∈ F_n} and F_n^+ : = max{x_+: x ∈ F_n}. Almost surely, there are for each n unique vectors λ_n ∈ F_n and τ_n ∈ F_n such that F_n^+ = (λ_n)_+ and F_n^- = (τ_n)_+; we refer to λ_n and τ_n as the leading and trailing points, respectively, of the frontier. Fill and Naiman provided rather sharp information about the typical and almost sure behavior of F^+, but somewhat crude information about F^-, namely, that for any ε > 0 and c_n → ∞ we have P(F_n^- - ln n ∈ (- (2 + ε) ln ln ln n, c_n)) → 1 (describing typical behavior) and almost surely limsup (F_n^- - ln n)/(ln ln n) ≤ 0 and liminf (F_n^- - ln n)/(ln ln ln n) ∈ [-2, -1]. In this extended abstract we use the theory of generators (minima of F_n) together with the first- and second-moment methods to improve considerably the trailing-point location results to F_n^- - (ln n - ln ln ln n) ⟶P -ln(d - 1) (describing typical behavior) and, for d ≥ 3, almost surely limsup [F_n^- -(ln n - ln ln ln n)] ≤ -ln(d - 2) + ln 2 and liminf [F_n^- -(ln n - ln ln ln n)] ≥ -ln d - ln 2.
Cite as

James Allen Fill, Daniel Q. Naiman, and Ao Sun. Sharpened Localization of the Trailing Point of the Pareto Record Frontier. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fill_et_al:LIPIcs.AofA.2024.28,
  author =	{Fill, James Allen and Naiman, Daniel Q. and Sun, Ao},
  title =	{{Sharpened Localization of the Trailing Point of the Pareto Record Frontier}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{28:1--28:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.28},
  URN =		{urn:nbn:de:0030-drops-204631},
  doi =		{10.4230/LIPIcs.AofA.2024.28},
  annote =	{Keywords: Multivariate records, Pareto records, generators, interior generators, minima, maxima, record-setting region, frontier, current records, boundary-crossing probabilities, first moment method, second moment method, orthants}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.54
NLTS Hamiltonians and Strongly-Explicit SoS Lower Bounds from Low-Rate Quantum LDPC Codes

Authors: Louis Golowich and Tali Kaufman

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
Recent constructions of the first asymptotically good quantum LDPC (qLDPC) codes led to two breakthroughs in complexity theory: the NLTS (No Low-Energy Trivial States) theorem (Anshu, Breuckmann, and Nirkhe, STOC'23), and explicit lower bounds against a linear number of levels of the Sum-of-Squares (SoS) hierarchy (Hopkins and Lin, FOCS'22). In this work, we obtain improvements to both of these results using qLDPC codes of low rate: - Whereas Anshu et al. only obtained NLTS Hamiltonians from qLDPC codes of linear dimension, we show the stronger result that qLDPC codes of arbitrarily small positive dimension yield NLTS Hamiltonians. - The SoS lower bounds of Hopkins and Lin are only weakly explicit because they require running Gaussian elimination to find a nontrivial codeword, which takes polynomial time. We resolve this shortcoming by introducing a new method of planting a strongly explicit nontrivial codeword in linear-distance qLDPC codes, which in turn yields strongly explicit SoS lower bounds. Our "planted" qLDPC codes may be of independent interest, as they provide a new way of ensuring a qLDPC code has positive dimension without resorting to parity check counting, and therefore provide more flexibility in the code construction.
Cite as

Louis Golowich and Tali Kaufman. NLTS Hamiltonians and Strongly-Explicit SoS Lower Bounds from Low-Rate Quantum LDPC Codes. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 54:1-54:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{golowich_et_al:LIPIcs.ITCS.2024.54,
  author =	{Golowich, Louis and Kaufman, Tali},
  title =	{{NLTS Hamiltonians and Strongly-Explicit SoS Lower Bounds from Low-Rate Quantum LDPC Codes}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{54:1--54:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.54},
  URN =		{urn:nbn:de:0030-drops-195829},
  doi =		{10.4230/LIPIcs.ITCS.2024.54},
  annote =	{Keywords: NLTS Hamiltonian, Quantum PCP, Sum-of-squares lower bound, Quantum LDPC code}
}
Document
Brief Announcement
DOI: 10.4230/LIPIcs.DISC.2022.42
Brief Announcement: Minimizing Congestion in Hybrid Demand-Aware Network Topologies

Authors: Wenkai Dai, Michael Dinitz, Klaus-Tycho Foerster, and Stefan Schmid

Published in: LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

Abstract
Emerging reconfigurable optical communication technologies enable demand-aware networks: networks whose static topology can be enhanced with demand-aware links optimized towards the traffic pattern the network serves. This paper studies the algorithmic problem of how to jointly optimize the topology and the routing in such demand-aware networks, to minimize congestion. We investigate this problem along two dimensions: (1) whether flows are splittable or unsplittable, and (2) whether routing on the hybrid topology is segregated or not, i.e., whether or not flows either have to use exclusively either the static network or the demand-aware connections. For splittable and segregated routing, we show that the problem is 2-approximable in general, but APX-hard even for uniform demands induced by a bipartite demand graph. For unsplittable and segregated routing, we show an upper bound of O(log m/ log log m) and a lower bound of Ω(log m/ log log m) for polynomial-time approximation algorithms, where m is the number of static links. Under splittable (resp., unsplittable) and non-segregated routing, even for demands of a single source (resp., destination), the problem cannot be approximated better than Ω(c_{max}/c_{min}) unless P=NP, where c_{max} (resp., c_{min}) denotes the maximum (resp., minimum) capacity. It is still NP-hard for uniform capacities, but can be solved efficiently for a single commodity and uniform capacities.
Cite as

Wenkai Dai, Michael Dinitz, Klaus-Tycho Foerster, and Stefan Schmid. Brief Announcement: Minimizing Congestion in Hybrid Demand-Aware Network Topologies. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 42:1-42:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dai_et_al:LIPIcs.DISC.2022.42,
  author =	{Dai, Wenkai and Dinitz, Michael and Foerster, Klaus-Tycho and Schmid, Stefan},
  title =	{{Brief Announcement: Minimizing Congestion in Hybrid Demand-Aware Network Topologies}},
  booktitle =	{36th International Symposium on Distributed Computing (DISC 2022)},
  pages =	{42:1--42:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-255-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{246},
  editor =	{Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.42},
  URN =		{urn:nbn:de:0030-drops-172330},
  doi =		{10.4230/LIPIcs.DISC.2022.42},
  annote =	{Keywords: Congestion, Reconfigurable Networks, Algorithms, Complexity}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.16
Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

Authors: Jason Gaitonde, Max Hopkins, Tali Kaufman, Shachar Lovett, and Ruizhe Zhang

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

Abstract
Fast mixing of random walks on hypergraphs (simplicial complexes) has recently led to myriad breakthroughs throughout theoretical computer science. Many important applications, however, (e.g. to LTCs, 2-2 games) rely on a more general class of underlying structures called posets, and crucially take advantage of non-simplicial structure. These works make it clear that the global expansion properties of posets depend strongly on their underlying architecture (e.g. simplicial, cubical, linear algebraic), but the overall phenomenon remains poorly understood. In this work, we quantify the advantage of different poset architectures in both a spectral and combinatorial sense, highlighting how regularity controls the spectral decay and edge-expansion of corresponding random walks. We show that the spectra of walks on expanding posets (Dikstein, Dinur, Filmus, Harsha APPROX-RANDOM 2018) concentrate in strips around a small number of approximate eigenvalues controlled by the regularity of the underlying poset. This gives a simple condition to identify poset architectures (e.g. the Grassmann) that exhibit strong (even exponential) decay of eigenvalues, versus architectures like hypergraphs whose eigenvalues decay linearly - a crucial distinction in applications to hardness of approximation and agreement testing such as the recent proof of the 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). We show these results lead to a tight characterization of edge-expansion on expanding posets in the 𝓁₂-regime (generalizing recent work of Bafna, Hopkins, Kaufman, and Lovett (SODA 2022)), and pay special attention to the case of the Grassmann where we show our results are tight for a natural set of sparsifications of the Grassmann graphs. We note for clarity that our results do not recover the characterization of expansion used in the proof of the 2-2 Games Conjecture which relies on 𝓁_∞ rather than 𝓁₂-structure.
Cite as

Jason Gaitonde, Max Hopkins, Tali Kaufman, Shachar Lovett, and Ruizhe Zhang. Eigenstripping, Spectral Decay, and Edge-Expansion on Posets. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 16:1-16:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gaitonde_et_al:LIPIcs.APPROX/RANDOM.2022.16,
  author =	{Gaitonde, Jason and Hopkins, Max and Kaufman, Tali and Lovett, Shachar and Zhang, Ruizhe},
  title =	{{Eigenstripping, Spectral Decay, and Edge-Expansion on Posets}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{16:1--16:24},
  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.2022.16},
  URN =		{urn:nbn:de:0030-drops-171381},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.16},
  annote =	{Keywords: High-dimensional expanders, posets, eposets}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.4
Pre-Constrained Encryption

Authors: Prabhanjan Ananth, Abhishek Jain, Zhengzhong Jin, and Giulio Malavolta

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

Abstract
In all existing encryption systems, the owner of the master secret key has the ability to decrypt all ciphertexts. In this work, we propose a new notion of pre-constrained encryption (PCE) where the owner of the master secret key does not have "full" decryption power. Instead, its decryption power is constrained in a pre-specified manner during the system setup. We present formal definitions and constructions of PCE, and discuss societal applications and implications to some well-studied cryptographic primitives.
Cite as

Prabhanjan Ananth, Abhishek Jain, Zhengzhong Jin, and Giulio Malavolta. Pre-Constrained Encryption. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ananth_et_al:LIPIcs.ITCS.2022.4,
  author =	{Ananth, Prabhanjan and Jain, Abhishek and Jin, Zhengzhong and Malavolta, Giulio},
  title =	{{Pre-Constrained Encryption}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{4:1--4:20},
  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.2022.4},
  URN =		{urn:nbn:de:0030-drops-156001},
  doi =		{10.4230/LIPIcs.ITCS.2022.4},
  annote =	{Keywords: Advanced encryption systems}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.11
Approximating the Norms of Graph Spanners

Authors: Eden Chlamtáč, Michael Dinitz, and Thomas Robinson

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
The l_p-norm of the degree vector was recently introduced by [Chlamtáč, Dinitz, Robinson ICALP '19] as a new cost metric for graph spanners, as it interpolates between two traditional notions of cost (the sparsity l_1 and the max degree l_infty) and is well-motivated from applications. We study this from an approximation algorithms point of view, analyzing old algorithms and designing new algorithms for this new context, as well as providing hardness results. Our main results are for the l_2-norm and stretch 3, where we give a tight analysis of the greedy algorithm and a new algorithm specifically tailored to this setting which gives an improved approximation ratio.
Cite as

Eden Chlamtáč, Michael Dinitz, and Thomas Robinson. Approximating the Norms of Graph Spanners. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 11:1-11:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chlamtac_et_al:LIPIcs.APPROX-RANDOM.2019.11,
  author =	{Chlamt\'{a}\v{c}, Eden and Dinitz, Michael and Robinson, Thomas},
  title =	{{Approximating the Norms of Graph Spanners}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{11:1--11:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.11},
  URN =		{urn:nbn:de:0030-drops-112261},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.11},
  annote =	{Keywords: Spanners, Approximations}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.8
Sherali - Adams Strikes Back

Authors: Ryan O'Donnell and Tselil Schramm

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

Abstract
Let G be any n-vertex graph whose random walk matrix has its nontrivial eigenvalues bounded in magnitude by 1/sqrt{Delta} (for example, a random graph G of average degree Theta(Delta) typically has this property). We show that the exp(c (log n)/(log Delta))-round Sherali - Adams linear programming hierarchy certifies that the maximum cut in such a G is at most 50.1 % (in fact, at most 1/2 + 2^{-Omega(c)}). For example, in random graphs with n^{1.01} edges, O(1) rounds suffice; in random graphs with n * polylog(n) edges, n^{O(1/log log n)} = n^{o(1)} rounds suffice. Our results stand in contrast to the conventional beliefs that linear programming hierarchies perform poorly for max-cut and other CSPs, and that eigenvalue/SDP methods are needed for effective refutation. Indeed, our results imply that constant-round Sherali - Adams can strongly refute random Boolean k-CSP instances with n^{ceil[k/2] + delta} constraints; previously this had only been done with spectral algorithms or the SOS SDP hierarchy.
Cite as

Ryan O'Donnell and Tselil Schramm. Sherali - Adams Strikes Back. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 8:1-8:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{odonnell_et_al:LIPIcs.CCC.2019.8,
  author =	{O'Donnell, Ryan and Schramm, Tselil},
  title =	{{Sherali - Adams Strikes Back}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{8:1--8:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.8},
  URN =		{urn:nbn:de:0030-drops-108309},
  doi =		{10.4230/LIPIcs.CCC.2019.8},
  annote =	{Keywords: Linear programming, Sherali, Adams, max-cut, graph eigenvalues, Sum-of-Squares}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.40
The Norms of Graph Spanners

Authors: Eden Chlamtáč, Michael Dinitz, and Thomas Robinson

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
A t-spanner of a graph G is a subgraph H in which all distances are preserved up to a multiplicative t factor. A classical result of Althöfer et al. is that for every integer k and every graph G, there is a (2k-1)-spanner of G with at most O(n^{1+1/k}) edges. But for some settings the more interesting notion is not the number of edges, but the degrees of the nodes. This spurred interest in and study of spanners with small maximum degree. However, this is not necessarily a robust enough objective: we would like spanners that not only have small maximum degree, but also have "few" nodes of "large" degree. To interpolate between these two extremes, in this paper we initiate the study of graph spanners with respect to the l_p-norm of their degree vector, thus simultaneously modeling the number of edges (the l_1-norm) and the maximum degree (the l_{infty}-norm). We give precise upper bounds for all ranges of p and stretch t: we prove that the greedy (2k-1)-spanner has l_p norm of at most max(O(n), O(n^{{k+p}/{kp}})), and that this bound is tight (assuming the Erdős girth conjecture). We also study universal lower bounds, allowing us to give "generic" guarantees on the approximation ratio of the greedy algorithm which generalize and interpolate between the known approximations for the l_1 and l_{infty} norm. Finally, we show that at least in some situations, the l_p norm behaves fundamentally differently from l_1 or l_{infty}: there are regimes (p=2 and stretch 3 in particular) where the greedy spanner has a provably superior approximation to the generic guarantee.
Cite as

Eden Chlamtáč, Michael Dinitz, and Thomas Robinson. The Norms of Graph Spanners. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chlamtac_et_al:LIPIcs.ICALP.2019.40,
  author =	{Chlamt\'{a}\v{c}, Eden and Dinitz, Michael and Robinson, Thomas},
  title =	{{The Norms of Graph Spanners}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{40:1--40:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.40},
  URN =		{urn:nbn:de:0030-drops-106163},
  doi =		{10.4230/LIPIcs.ICALP.2019.40},
  annote =	{Keywords: spanners, approximations}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.31
Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere

Authors: Vijay Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee

Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

Abstract
For an n-variate order-d tensor A, define A_{max} := sup_{||x||_2 = 1} <A,x^(otimes d)>, to be the maximum value taken by the tensor on the unit sphere. It is known that for a random tensor with i.i.d. +1/-1 entries, A_{max} <= sqrt(n.d.log(d)) w.h.p. We study the problem of efficiently certifying upper bounds on A_{max} via the natural relaxation from the Sum of Squares (SoS) hierarchy. Our results include: * When A is a random order-q tensor, we prove that q levels of SoS certifies an upper bound B on A_{max} that satisfies B <= A_{max} * (n/q^(1-o(1)))^(q/4-1/2) w.h.p. Our upper bound improves a result of Montanari and Richard (NIPS 2014) when q is large. * We show the above bound is the best possible up to lower order terms, namely the optimum of the level-q SoS relaxation is at least A_{max} * (n/q^(1+o(1)))^(q/4-1/2). * When A is a random order-d tensor, we prove that q levels of SoS certifies an upper bound B on A_{max} that satisfies B <= A_{max} * (n*polylog/q)^(d/4 - 1/2) w.h.p. For growing q, this improves upon the bound certified by constant levels of SoS. This answers in part, a question posed by Hopkins, Shi, and Steurer (COLT 2015), who tightly characterized constant levels of SoS.
Cite as

Vijay Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee. Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 31:1-31:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bhattiprolu_et_al:LIPIcs.APPROX-RANDOM.2017.31,
  author =	{Bhattiprolu, Vijay and Guruswami, Venkatesan and Lee, Euiwoong},
  title =	{{Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{31:1--31:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.31},
  URN =		{urn:nbn:de:0030-drops-75808},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.31},
  annote =	{Keywords: Sum-of-Squares, Optimization over Sphere, Random Polynomials}
}
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