12 Search Results for "Hopkins, Max"

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.CCC.2024.3
Solving Unique Games over Globally Hypercontractive Graphs

Authors: Mitali Bafna and Dor Minzer

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

Abstract
We study the complexity of affine Unique-Games (UG) over globally hypercontractive graphs, which are graphs that are not small set expanders but admit a useful and succinct characterization of all small sets that violate the small-set expansion property. This class of graphs includes the Johnson and Grassmann graphs, which have played a pivotal role in recent PCP constructions for UG, and their generalizations via high-dimensional expanders. We show new rounding techniques for higher degree sum-of-squares (SoS) relaxations for worst-case optimization. In particular, our algorithm shows how to round "low-entropy" pseudodistributions, broadly extending the algorithmic framework of [Mitali Bafna et al., 2021]. At a high level, [Mitali Bafna et al., 2021] showed how to round pseudodistributions for problems where there is a "unique" good solution. We extend their framework by exhibiting a rounding for problems where there might be "few good solutions". Our result suggests that UG is easy on globally hypercontractive graphs, and therefore highlights the importance of graphs that lack such a characterization in the context of PCP reductions for UG.
Cite as

Mitali Bafna and Dor Minzer. Solving Unique Games over Globally Hypercontractive Graphs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bafna_et_al:LIPIcs.CCC.2024.3,
  author =	{Bafna, Mitali and Minzer, Dor},
  title =	{{Solving Unique Games over Globally Hypercontractive Graphs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{3:1--3:15},
  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.2024.3},
  URN =		{urn:nbn:de:0030-drops-203996},
  doi =		{10.4230/LIPIcs.CCC.2024.3},
  annote =	{Keywords: unique games, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.10
Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs

Authors: Xin Li and Yan Zhong

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

Abstract
Affine extractors give some of the best-known lower bounds for various computational models, such as AC⁰ circuits, parity decision trees, and general Boolean circuits. However, they are not known to give strong lower bounds for read-once branching programs (ROBPs). In a recent work, Gryaznov, Pudlák, and Talebanfard (CCC' 22) introduced a stronger version of affine extractors known as directional affine extractors, together with a generalization of ROBPs where each node can make linear queries, and showed that the former implies strong lower bound for a certain type of the latter known as strongly read-once linear branching programs (SROLBPs). Their main result gives explicit constructions of directional affine extractors for entropy k > 2n/3, which implies average-case complexity 2^{n/3-o(n)} against SROLBPs with exponentially small correlation. A follow-up work by Chattopadhyay and Liao (CCC' 23) improves the hardness to 2^{n-o(n)} at the price of increasing the correlation to polynomially large, via a new connection to sumset extractors introduced by Chattopadhyay and Li (STOC' 16) and explicit constructions of such extractors by Chattopadhyay and Liao (STOC' 22). Both works left open the questions of better constructions of directional affine extractors and improved average-case complexity against SROLBPs in the regime of small correlation. This paper provides a much more in-depth study of directional affine extractors, SROLBPs, and ROBPs. Our main results include: - An explicit construction of directional affine extractors with k = o(n) and exponentially small error, which gives average-case complexity 2^{n-o(n)} against SROLBPs with exponentially small correlation, thus answering the two open questions raised in previous works. - An explicit function in AC⁰ that gives average-case complexity 2^{(1-δ)n} against ROBPs with negligible correlation, for any constant δ > 0. Previously, no such average-case hardness is known, and the best size lower bound for any function in AC⁰ against ROBPs is 2^Ω(n). One of the key ingredients in our constructions is a new linear somewhere condenser for affine sources, which is based on dimension expanders. The condenser also leads to an unconditional improvement of the entropy requirement of explicit affine extractors with negligible error. We further show that the condenser also works for general weak random sources, under the Polynomial Freiman-Ruzsa Theorem in 𝖥₂ⁿ, recently proved by Gowers, Green, Manners, and Tao (arXiv' 23).
Cite as

Xin Li and Yan Zhong. Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{li_et_al:LIPIcs.CCC.2024.10,
  author =	{Li, Xin and Zhong, Yan},
  title =	{{Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{10:1--10:14},
  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.2024.10},
  URN =		{urn:nbn:de:0030-drops-204060},
  doi =		{10.4230/LIPIcs.CCC.2024.10},
  annote =	{Keywords: Randomness Extractors, Affine, Read-once Linear Branching Programs, Low-degree polynomials, AC⁰ circuits}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.117
Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle

Authors: Aaron Potechin and Aaron Zhang

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

Abstract
We show that the minimum total coefficient size of a Nullstellensatz proof of the pigeonhole principle on n+1 pigeons and n holes is 2^{Θ(n)}. We also investigate the ordering principle and construct an explicit Nullstellensatz proof for the ordering principle on n elements with total coefficient size 2ⁿ - n.
Cite as

Aaron Potechin and Aaron Zhang. Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 117:1-117:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{potechin_et_al:LIPIcs.ICALP.2024.117,
  author =	{Potechin, Aaron and Zhang, Aaron},
  title =	{{Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{117:1--117:20},
  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.2024.117},
  URN =		{urn:nbn:de:0030-drops-202604},
  doi =		{10.4230/LIPIcs.ICALP.2024.117},
  annote =	{Keywords: Proof complexity, Nullstellensatz, pigeonhole principle, coefficient size}
}
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.
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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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