8 Search Results for "Hopkins, Max"


Document
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
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
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
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
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
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
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
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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