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LIPIcs, Volume 169
35th Computational Complexity Conference (CCC 2020)
-
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Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Computational Complexity Conference (CCC)
Publication Details
- published at: 2020-07-17
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-156-6
- DBLP: db/conf/coco/coco2020
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Complete Volume
LIPIcs, Volume 169, CCC 2020, Complete Volume
Abstract
LIPIcs, Volume 169, CCC 2020, Complete Volume
Cite as
35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 1-1068, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@Proceedings{saraf:LIPIcs.CCC.2020,
title = {{LIPIcs, Volume 169, CCC 2020, Complete Volume}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {1--1068},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020},
URN = {urn:nbn:de:0030-drops-125518},
doi = {10.4230/LIPIcs.CCC.2020},
annote = {Keywords: LIPIcs, Volume 169, CCC 2020, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{saraf:LIPIcs.CCC.2020.0,
author = {Saraf, Shubhangi},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {0:i--0:xvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.0},
URN = {urn:nbn:de:0030-drops-125520},
doi = {10.4230/LIPIcs.CCC.2020.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Near-Optimal Erasure List-Decodable Codes
Abstract
A code 𝒞 ⊆ {0,1}^n̅ is (s,L) erasure list-decodable if for every word w, after erasing any s symbols of w, the remaining n̅-s symbols have at most L possible completions into a codeword of 𝒞. Non-explicitly, there exist binary ((1-τ)n̅,L) erasure list-decodable codes with rate approaching τ and tiny list-size L = O(log 1/(τ)). Achieving either of these parameters explicitly is a natural open problem (see, e.g., [Guruswami and Indyk, 2002; Guruswami, 2003; Guruswami, 2004]). While partial progress on the problem has been achieved, no prior nontrivial explicit construction achieved rate better than Ω(τ²) or list-size smaller than Ω(1/τ). Furthermore, Guruswami showed no linear code can have list-size smaller than Ω(1/τ) [Guruswami, 2003]. We construct an explicit binary ((1-τ)n̅,L) erasure list-decodable code having rate τ^(1+γ) (for any constant γ > 0 and small τ) and list-size poly(log 1/τ), answering simultaneously both questions, and exhibiting an explicit non-linear code that provably beats the best possible linear code.
The binary erasure list-decoding problem is equivalent to the construction of explicit, low-error, strong dispersers outputting one bit with minimal entropy-loss and seed-length. For error ε, no prior explicit construction achieved seed-length better than 2log(1/ε) or entropy-loss smaller than 2log(1/ε), which are the best possible parameters for extractors. We explicitly construct an ε-error one-bit strong disperser with near-optimal seed-length (1+γ)log(1/ε) and entropy-loss O(log log1/ε).
The main ingredient in our construction is a new (and almost-optimal) unbalanced two-source extractor. The extractor extracts one bit with constant error from two independent sources, where one source has length n and tiny min-entropy O(log log n) and the other source has length O(log n) and arbitrarily small constant min-entropy rate. When instantiated as a balanced two-source extractor, it improves upon Raz’s extractor [Raz, 2005] in the constant error regime. The construction incorporates recent components and ideas from extractor theory with a delicate and novel analysis needed in order to solve dependency and error issues that prevented previous papers (such as [Li, 2015; Chattopadhyay and Zuckerman, 2019; Cohen, 2016]) from achieving the above results.
Cite as
Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma. Near-Optimal Erasure List-Decodable Codes. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 1:1-1:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{benaroya_et_al:LIPIcs.CCC.2020.1,
author = {Ben-Aroya, Avraham and Doron, Dean and Ta-Shma, Amnon},
title = {{Near-Optimal Erasure List-Decodable Codes}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {1:1--1:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.1},
URN = {urn:nbn:de:0030-drops-125531},
doi = {10.4230/LIPIcs.CCC.2020.1},
annote = {Keywords: Dispersers, Erasure codes, List decoding, Ramsey graphs, Two-source extractors}
}
Document
A Quadratic Lower Bound for Algebraic Branching Programs
Abstract
We show that any Algebraic Branching Program (ABP) computing the polynomial ∑_{i=1}^n xⁿ_i has at least Ω(n²) vertices. This improves upon the lower bound of Ω(nlog n), which follows from the classical result of Baur and Strassen [Volker Strassen, 1973; Walter Baur and Volker Strassen, 1983], and extends the results of Kumar [Mrinal Kumar, 2019], which showed a quadratic lower bound for homogeneous ABPs computing the same polynomial.
Our proof relies on a notion of depth reduction which is reminiscent of similar statements in the context of matrix rigidity, and shows that any small enough ABP computing the polynomial ∑_{i=1}^n xⁿ_i can be depth reduced to essentially a homogeneous ABP of the same size which computes the polynomial ∑_{i=1}^n xⁿ_i + ε(𝐱), for a structured "error polynomial" ε(𝐱). To complete the proof, we then observe that the lower bound in [Mrinal Kumar, 2019] is robust enough and continues to hold for all polynomials ∑_{i=1}^n xⁿ_i + ε(𝐱), where ε(𝐱) has the appropriate structure.
Cite as
Prerona Chatterjee, Mrinal Kumar, Adrian She, and Ben Lee Volk. A Quadratic Lower Bound for Algebraic Branching Programs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2020.2,
author = {Chatterjee, Prerona and Kumar, Mrinal and She, Adrian and Volk, Ben Lee},
title = {{A Quadratic Lower Bound for Algebraic Branching Programs}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {2:1--2:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.2},
URN = {urn:nbn:de:0030-drops-125546},
doi = {10.4230/LIPIcs.CCC.2020.2},
annote = {Keywords: Algebraic Branching Programs, Lower Bound}
}
Document
Super Strong ETH Is True for PPSZ with Small Resolution Width
Abstract
We construct k-CNFs with m variables on which the strong version of PPSZ k-SAT algorithm, which uses resolution of width bounded by O(√{log log m}), has success probability at most 2^{-(1-(1 + ε)2/k)m} for every ε > 0. Previously such a bound was known only for the weak PPSZ algorithm which exhaustively searches through small subformulas of the CNF to see if any of them forces the value of a given variable, and for strong PPSZ the best known previous upper bound was 2^{-(1-O(log(k)/k))m} (Pudlák et al., ICALP 2017).
Cite as
Dominik Scheder and Navid Talebanfard. Super Strong ETH Is True for PPSZ with Small Resolution Width. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 3:1-3:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{scheder_et_al:LIPIcs.CCC.2020.3,
author = {Scheder, Dominik and Talebanfard, Navid},
title = {{Super Strong ETH Is True for PPSZ with Small Resolution Width}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {3:1--3:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.3},
URN = {urn:nbn:de:0030-drops-125552},
doi = {10.4230/LIPIcs.CCC.2020.3},
annote = {Keywords: k-SAT, PPSZ, Resolution}
}
Document
Approximability of the Eight-Vertex Model
Abstract
We initiate a study of the classification of approximation complexity of the eight-vertex model defined over 4-regular graphs. The eight-vertex model, together with its special case the six-vertex model, is one of the most extensively studied models in statistical physics, and can be stated as a problem of counting weighted orientations in graph theory. Our result concerns the approximability of the partition function on all 4-regular graphs, classified according to the parameters of the model. Our complexity results conform to the phase transition phenomenon from physics.
We introduce a quantum decomposition of the eight-vertex model and prove a set of closure properties in various regions of the parameter space. Furthermore, we show that there are extra closure properties on 4-regular planar graphs. These regions of the parameter space are concordant with the phase transition threshold. Using these closure properties, we derive polynomial time approximation algorithms via Markov chain Monte Carlo. We also show that the eight-vertex model is NP-hard to approximate on the other side of the phase transition threshold.
Cite as
Jin-Yi Cai, Tianyu Liu, Pinyan Lu, and Jing Yu. Approximability of the Eight-Vertex Model. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cai_et_al:LIPIcs.CCC.2020.4,
author = {Cai, Jin-Yi and Liu, Tianyu and Lu, Pinyan and Yu, Jing},
title = {{Approximability of the Eight-Vertex Model}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {4:1--4:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.4},
URN = {urn:nbn:de:0030-drops-125564},
doi = {10.4230/LIPIcs.CCC.2020.4},
annote = {Keywords: Approximate complexity, the eight-vertex model, Markov chain Monte Carlo}
}
Document
Lower Bounds for Matrix Factorization
Abstract
We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various incarnations in computer science; the most significant being in the context of lower bounds for algebraic circuits which compute linear transformations, matrix rigidity and data structure lower bounds.
We first show, for every constant d, a deterministic construction in time exp(n^(1-Ω(1/d))) of a family {M_n} of n × n matrices which cannot be expressed as a product M_n = A_1 ⋯ A_d where the total sparsity of A_1,…,A_d is less than n^(1+1/(2d)). In other words, any depth-d linear circuit computing the linear transformation M_n⋅ 𝐱 has size at least n^(1+Ω(1/d)). This improves upon the prior best lower bounds for this problem, which are barely super-linear, and were obtained by a long line of research based on the study of super-concentrators (albeit at the cost of a blow up in the time required to construct these matrices).
We then outline an approach for proving improved lower bounds through a certain derandomization problem, and use this approach to prove asymptotically optimal quadratic lower bounds for natural special cases, which generalize many of the common matrix decompositions.
Cite as
Mrinal Kumar and Ben Lee Volk. Lower Bounds for Matrix Factorization. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kumar_et_al:LIPIcs.CCC.2020.5,
author = {Kumar, Mrinal and Volk, Ben Lee},
title = {{Lower Bounds for Matrix Factorization}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {5:1--5:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.5},
URN = {urn:nbn:de:0030-drops-125578},
doi = {10.4230/LIPIcs.CCC.2020.5},
annote = {Keywords: Algebraic Complexity, Linear Circuits, Matrix Factorization, Lower Bounds}
}
Document
Log-Seed Pseudorandom Generators via Iterated Restrictions
Abstract
There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The "iterated restrictions" approach, pioneered by Ajtai and Wigderson [Ajtai and Wigderson, 1989], has provided PRGs with seed length polylog n or even Õ(log n) for several restricted models of computation. Can this approach ever achieve the optimal seed length of O(log n)?
In this work, we answer this question in the affirmative. Using the iterated restrictions approach, we construct an explicit PRG for read-once depth-2 AC⁰[⊕] formulas with seed length O(log n) + Õ(log(1/ε)). In particular, we achieve optimal seed length O(log n) with near-optimal error ε = exp(-Ω̃(log n)). Even for constant error, the best prior PRG for this model (which includes read-once CNFs and read-once 𝔽₂-polynomials) has seed length Θ(log n ⋅ (log log n)²) [Chin Ho Lee, 2019].
A key step in the analysis of our PRG is a tail bound for subset-wise symmetric polynomials, a generalization of elementary symmetric polynomials. Like elementary symmetric polynomials, subset-wise symmetric polynomials provide a way to organize the expansion of ∏_{i=1}^m (1 + y_i). Elementary symmetric polynomials simply organize the terms by degree, i.e., they keep track of the number of variables participating in each monomial. Subset-wise symmetric polynomials keep track of more data: for a fixed partition of [m], they keep track of the number of variables from each subset participating in each monomial. Our tail bound extends prior work by Gopalan and Yehudayoff [Gopalan and Yehudayoff, 2014] on elementary symmetric polynomials.
Cite as
Dean Doron, Pooya Hatami, and William M. Hoza. Log-Seed Pseudorandom Generators via Iterated Restrictions. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 6:1-6:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{doron_et_al:LIPIcs.CCC.2020.6,
author = {Doron, Dean and Hatami, Pooya and Hoza, William M.},
title = {{Log-Seed Pseudorandom Generators via Iterated Restrictions}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {6:1--6:36},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.6},
URN = {urn:nbn:de:0030-drops-125586},
doi = {10.4230/LIPIcs.CCC.2020.6},
annote = {Keywords: Pseudorandom generators, Pseudorandom restrictions, Read-once depth-2 formulas, Parity gates}
}
Document
Quantum Lower Bounds for Approximate Counting via Laurent Polynomials
Abstract
We study quantum algorithms that are given access to trusted and untrusted quantum witnesses. We establish strong limitations of such algorithms, via new techniques based on Laurent polynomials (i.e., polynomials with positive and negative integer exponents). Specifically, we resolve the complexity of approximate counting, the problem of multiplicatively estimating the size of a nonempty set S ⊆ [N], in two natural generalizations of quantum query complexity.
Our first result holds in the standard Quantum Merlin - Arthur (QMA) setting, in which a quantum algorithm receives an untrusted quantum witness. We show that, if the algorithm makes T quantum queries to S, and also receives an (untrusted) m-qubit quantum witness, then either m = Ω(|S|) or T = Ω(√{N/|S|}). This is optimal, matching the straightforward protocols where the witness is either empty, or specifies all the elements of S. As a corollary, this resolves the open problem of giving an oracle separation between SBP, the complexity class that captures approximate counting, and QMA.
In our second result, we ask what if, in addition to a membership oracle for S, a quantum algorithm is also given "QSamples" - i.e., copies of the state |S⟩ = 1/√|S| ∑_{i ∈ S} |i⟩ - or even access to a unitary transformation that enables QSampling? We show that, even then, the algorithm needs either Θ(√{N/|S|}) queries or else Θ(min{|S|^{1/3},√{N/|S|}}) QSamples or accesses to the unitary.
Our lower bounds in both settings make essential use of Laurent polynomials, but in different ways.
Cite as
Scott Aaronson, Robin Kothari, William Kretschmer, and Justin Thaler. Quantum Lower Bounds for Approximate Counting via Laurent Polynomials. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 7:1-7:47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{aaronson_et_al:LIPIcs.CCC.2020.7,
author = {Aaronson, Scott and Kothari, Robin and Kretschmer, William and Thaler, Justin},
title = {{Quantum Lower Bounds for Approximate Counting via Laurent Polynomials}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {7:1--7:47},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.7},
URN = {urn:nbn:de:0030-drops-125593},
doi = {10.4230/LIPIcs.CCC.2020.7},
annote = {Keywords: Approximate counting, Laurent polynomials, QSampling, query complexity}
}
Document
A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials
Abstract
In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of Σ^{[3]}ΠΣΠ^{[2]} circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials 𝒬 satisfy that for every two polynomials Q₁,Q₂ ∈ 𝒬 there is a subset 𝒦 ⊂ 𝒬, such that Q₁,Q₂ ∉ 𝒦 and whenever Q₁ and Q₂ vanish then ∏_{Q_i∈𝒦} Q_i vanishes, then the linear span of the polynomials in 𝒬 has dimension O(1). This extends the earlier result [Amir Shpilka, 2019] that showed a similar conclusion when |𝒦| = 1.
An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generated by the two quadratics. This step extends a result from [Amir Shpilka, 2019] that studied the case when one quadratic polynomial is in the radical of two other quadratics.
Cite as
Shir Peleg and Amir Shpilka. A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 8:1-8:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{peleg_et_al:LIPIcs.CCC.2020.8,
author = {Peleg, Shir and Shpilka, Amir},
title = {{A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {8:1--8:33},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.8},
URN = {urn:nbn:de:0030-drops-125606},
doi = {10.4230/LIPIcs.CCC.2020.8},
annote = {Keywords: Algebraic computation, Computational complexity, Computational geometry}
}
Document
Simultaneous Max-Cut Is Harder to Approximate Than Max-Cut
Abstract
A systematic study of simultaneous optimization of constraint satisfaction problems was initiated by Bhangale et al. [ICALP, 2015]. The simplest such problem is the simultaneous Max-Cut. Bhangale et al. [SODA, 2018] gave a .878-minimum approximation algorithm for simultaneous Max-Cut which is almost optimal assuming the Unique Games Conjecture (UGC). For single instance Max-Cut, Goemans-Williamson [JACM, 1995] gave an α_GW-approximation algorithm where α_GW ≈ .87856720... which is optimal assuming the UGC.
It was left open whether one can achieve an α_GW-minimum approximation algorithm for simultaneous Max-Cut. We answer the question by showing that there exists an absolute constant ε₀ ≥ 10^{-5} such that it is NP-hard to get an (α_GW- ε₀)-minimum approximation for simultaneous Max-Cut assuming the Unique Games Conjecture.
Cite as
Amey Bhangale and Subhash Khot. Simultaneous Max-Cut Is Harder to Approximate Than Max-Cut. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bhangale_et_al:LIPIcs.CCC.2020.9,
author = {Bhangale, Amey and Khot, Subhash},
title = {{Simultaneous Max-Cut Is Harder to Approximate Than Max-Cut}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {9:1--9:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.9},
URN = {urn:nbn:de:0030-drops-125610},
doi = {10.4230/LIPIcs.CCC.2020.9},
annote = {Keywords: Simultaneous CSPs, Unique Games hardness, Max-Cut}
}
Document
Hitting Sets Give Two-Sided Derandomization of Small Space
Abstract
A hitting set is a "one-sided" variant of a pseudorandom generator (PRG), naturally suited to derandomizing algorithms that have one-sided error. We study the problem of using a given hitting set to derandomize algorithms that have two-sided error, focusing on space-bounded algorithms. For our first result, we show that if there is a log-space hitting set for polynomial-width read-once branching programs (ROBPs), then not only does 𝐋 = 𝐑𝐋, but 𝐋 = 𝐁𝐏𝐋 as well. This answers a question raised by Hoza and Zuckerman [William M. Hoza and David Zuckerman, 2018].
Next, we consider constant-width ROBPs. We show that if there are log-space hitting sets for constant-width ROBPs, then given black-box access to a constant-width ROBP f, it is possible to deterministically estimate 𝔼[f] to within ± ε in space O(log(n/ε)). Unconditionally, we give a deterministic algorithm for this problem with space complexity O(log² n + log(1/ε)), slightly improving over previous work.
Finally, we investigate the limits of this line of work. Perhaps the strongest reduction along these lines one could hope for would say that for every explicit hitting set, there is an explicit PRG with similar parameters. In the setting of constant-width ROBPs over a large alphabet, we prove that establishing such a strong reduction is at least as difficult as constructing a good PRG outright. Quantitatively, we prove that if the strong reduction holds, then for every constant α > 0, there is an explicit PRG for constant-width ROBPs with seed length O(log^{1 + α} n). Along the way, unconditionally, we construct an improved hitting set for ROBPs over a large alphabet.
Cite as
Kuan Cheng and William M. Hoza. Hitting Sets Give Two-Sided Derandomization of Small Space. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 10:1-10:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cheng_et_al:LIPIcs.CCC.2020.10,
author = {Cheng, Kuan and Hoza, William M.},
title = {{Hitting Sets Give Two-Sided Derandomization of Small Space}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {10:1--10:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.10},
URN = {urn:nbn:de:0030-drops-125625},
doi = {10.4230/LIPIcs.CCC.2020.10},
annote = {Keywords: hitting sets, derandomization, read-once branching programs}
}
Document
Palette-Alternating Tree Codes
Abstract
A tree code is an edge-coloring of the complete infinite binary tree such that every two nodes of equal depth have a fraction - bounded away from 0 - of mismatched colors between the corresponding paths to their least common ancestor. Tree codes were introduced in a seminal work by Schulman [Schulman, 1993] and serve as a key ingredient in almost all deterministic interactive coding schemes. The number of colors effects the coding scheme’s rate.
It is shown that 4 is precisely the least number of colors for which tree codes exist. Thus, tree-code-based coding schemes cannot achieve rate larger than 1/2. To overcome this barrier, a relaxed notion called palette-alternating tree codes is introduced, in which the number of colors can depend on the layer. We prove the existence of such constructs in which most layers use 2 colors - the bare minimum. The distance-rate tradeoff we obtain matches the Gilbert-Varshamov bound.
Based on palette-alternating tree codes, we devise a deterministic interactive coding scheme against adversarial errors that approaches capacity. To analyze our protocol, we prove a structural result on the location of failed communication-rounds induced by the error pattern enforced by the adversary. Our coding scheme is efficient given an explicit palette-alternating tree code and serves as an alternative to the scheme obtained by [R. Gelles et al., 2016].
Cite as
Gil Cohen and Shahar Samocha. Palette-Alternating Tree Codes. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 11:1-11:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cohen_et_al:LIPIcs.CCC.2020.11,
author = {Cohen, Gil and Samocha, Shahar},
title = {{Palette-Alternating Tree Codes}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {11:1--11:29},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.11},
URN = {urn:nbn:de:0030-drops-125632},
doi = {10.4230/LIPIcs.CCC.2020.11},
annote = {Keywords: Tree Codes, Coding Theory, Interactive Coding Scheme}
}
Document
Search Problems in Algebraic Complexity, GCT, and Hardness of Generators for Invariant Rings
Abstract
We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of SL_n(ℂ)-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions).
We develop a general framework, denoted algebraic circuit search problems, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computing succinct encodings for generators for invariant rings.
Cite as
Ankit Garg, Christian Ikenmeyer, Visu Makam, Rafael Oliveira, Michael Walter, and Avi Wigderson. Search Problems in Algebraic Complexity, GCT, and Hardness of Generators for Invariant Rings. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{garg_et_al:LIPIcs.CCC.2020.12,
author = {Garg, Ankit and Ikenmeyer, Christian and Makam, Visu and Oliveira, Rafael and Walter, Michael and Wigderson, Avi},
title = {{Search Problems in Algebraic Complexity, GCT, and Hardness of Generators for Invariant Rings}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {12:1--12:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.12},
URN = {urn:nbn:de:0030-drops-125645},
doi = {10.4230/LIPIcs.CCC.2020.12},
annote = {Keywords: generators for invariant rings, succinct encodings}
}
Document
Statistical Physics Approaches to Unique Games
Abstract
We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the "yes" case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the "yes" case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would be strong negative evidence for the truth of the Unique Games Conjecture.
Cite as
Matthew Coulson, Ewan Davies, Alexandra Kolla, Viresh Patel, and Guus Regts. Statistical Physics Approaches to Unique Games. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 13:1-13:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{coulson_et_al:LIPIcs.CCC.2020.13,
author = {Coulson, Matthew and Davies, Ewan and Kolla, Alexandra and Patel, Viresh and Regts, Guus},
title = {{Statistical Physics Approaches to Unique Games}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {13:1--13:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.13},
URN = {urn:nbn:de:0030-drops-125650},
doi = {10.4230/LIPIcs.CCC.2020.13},
annote = {Keywords: Unique Games Conjecture, approximation algorithm, Potts model, cluster expansion, polynomial interpolation}
}
Document
Schur Polynomials Do Not Have Small Formulas If the Determinant Doesn't
Abstract
Schur Polynomials are families of symmetric polynomials that have been classically studied in Combinatorics and Algebra alike. They play a central role in the study of Symmetric functions, in Representation theory [Stanley, 1999], in Schubert calculus [Ledoux and Malham, 2010] as well as in Enumerative combinatorics [Gasharov, 1996; Stanley, 1984; Stanley, 1999]. In recent years, they have also shown up in various incarnations in Computer Science, e.g, Quantum computation [Hallgren et al., 2000; Ryan O'Donnell and John Wright, 2015] and Geometric complexity theory [Ikenmeyer and Panova, 2017].
However, unlike some other families of symmetric polynomials like the Elementary Symmetric polynomials, the Power Symmetric polynomials and the Complete Homogeneous Symmetric polynomials, the computational complexity of syntactically computing Schur polynomials has not been studied much. In particular, it is not known whether Schur polynomials can be computed efficiently by algebraic formulas. In this work, we address this question, and show that unless every polynomial with a small algebraic branching program (ABP) has a small algebraic formula, there are Schur polynomials that cannot be computed by algebraic formula of polynomial size. In other words, unless the algebraic complexity class VBP is equal to the complexity class VF, there exist Schur polynomials which do not have polynomial size algebraic formulas.
As a consequence of our proof, we also show that computing the determinant of certain generalized Vandermonde matrices is essentially as hard as computing the general symbolic determinant. To the best of our knowledge, these are one of the first hardness results of this kind for families of polynomials which are not multilinear. A key ingredient of our proof is the study of composition of well behaved algebraically independent polynomials with a homogeneous polynomial, and might be of independent interest.
Cite as
Prasad Chaugule, Mrinal Kumar, Nutan Limaye, Chandra Kanta Mohapatra, Adrian She, and Srikanth Srinivasan. Schur Polynomials Do Not Have Small Formulas If the Determinant Doesn't. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 14:1-14:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{chaugule_et_al:LIPIcs.CCC.2020.14,
author = {Chaugule, Prasad and Kumar, Mrinal and Limaye, Nutan and Mohapatra, Chandra Kanta and She, Adrian and Srinivasan, Srikanth},
title = {{Schur Polynomials Do Not Have Small Formulas If the Determinant Doesn't}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {14:1--14:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.14},
URN = {urn:nbn:de:0030-drops-125660},
doi = {10.4230/LIPIcs.CCC.2020.14},
annote = {Keywords: Schur polynomial, Jacobian, Algebraic independence, Generalized Vandermonde determinant, Taylor expansion, Formula complexity, Lower bound}
}
Document
Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates
Abstract
The class 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[s]∘𝒢 consists of Boolean functions computable by size-s de Morgan formulas whose leaves are any Boolean functions from a class 𝒢. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n^{1.99}]∘𝒢, for classes 𝒢 of functions with low communication complexity. Let R^(k)(𝒢) be the maximum k-party number-on-forehead randomized communication complexity of a function in 𝒢. Among other results, we show that:
- The Generalized Inner Product function 𝖦𝖨𝖯^k_n cannot be computed in 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[s]∘𝒢 on more than 1/2+ε fraction of inputs for s = o(n²/{(k⋅4^k⋅R^(k)(𝒢)⋅log (n/ε)⋅log(1/ε))²}). This significantly extends the lower bounds against bipartite formulas obtained by [Avishay Tal, 2017]. As a corollary, we get an average-case lower bound for 𝖦𝖨𝖯^k_n against 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^{1.99}]∘𝖯𝖳𝖥^{k-1}, i.e., sub-quadratic-size de Morgan formulas with degree-(k-1) PTF (polynomial threshold function) gates at the bottom.
- There is a PRG of seed length n/2 + O(√s⋅R^(2)(𝒢)⋅log(s/ε)⋅log(1/ε)) that ε-fools FORMULA[s]∘𝒢. For the special case of FORMULA[s]∘𝖫𝖳𝖥, i.e., size-s formulas with LTF (linear threshold function) gates at the bottom, we get the better seed length O(n^{1/2}⋅s^{1/4}⋅log(n)⋅log(n/ε)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n half-spaces in the regime where ε ≤ 1/n, complementing a recent result of [Ryan O'Donnell et al., 2019].
- There exists a randomized 2^{n-t}-time #SAT algorithm for 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[s]∘𝒢, where t = Ω(n/{√s⋅log²(s)⋅R^(2)(𝒢)})^{1/2}. In particular, this implies a nontrivial #SAT algorithm for 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^1.99]∘𝖫𝖳𝖥.
- The Minimum Circuit Size Problem is not in 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^1.99]∘𝖷𝖮𝖱; thereby making progress on hardness magnification, in connection with results from [Igor Carboni Oliveira et al., 2019; Lijie Chen et al., 2019]. On the algorithmic side, we show that the concept class 𝖥𝖮𝖱𝖬𝖴𝖫𝖠[n^1.99]∘𝖷𝖮𝖱 can be PAC-learned in time 2^O(n/log n).
Cite as
Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, and Igor C. Oliveira. Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 15:1-15:41, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kabanets_et_al:LIPIcs.CCC.2020.15,
author = {Kabanets, Valentine and Koroth, Sajin and Lu, Zhenjian and Myrisiotis, Dimitrios and Oliveira, Igor C.},
title = {{Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {15:1--15:41},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.15},
URN = {urn:nbn:de:0030-drops-125673},
doi = {10.4230/LIPIcs.CCC.2020.15},
annote = {Keywords: de Morgan formulas, circuit lower bounds, satisfiability (SAT), pseudorandom generators (PRGs), learning, communication complexity, polynomial threshold functions (PTFs), parities}
}
Document
On the Quantum Complexity of Closest Pair and Related Problems
Abstract
The closest pair problem is a fundamental problem of computational geometry: given a set of n points in a d-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this problem in O(n log n) time in constant dimensions (i.e., when d = O(1)). This paper asks and answers the question of the problem’s quantum time complexity. Specifically, we give an Õ(n^(2/3)) algorithm in constant dimensions, which is optimal up to a polylogarithmic factor by the lower bound on the quantum query complexity of element distinctness. The key to our algorithm is an efficient history-independent data structure that supports quantum interference.
In polylog(n) dimensions, no known quantum algorithms perform better than brute force search, with a quadratic speedup provided by Grover’s algorithm. To give evidence that the quadratic speedup is nearly optimal, we initiate the study of quantum fine-grained complexity and introduce the Quantum Strong Exponential Time Hypothesis (QSETH), which is based on the assumption that Grover’s algorithm is optimal for CNF-SAT when the clause width is large. We show that the naïve Grover approach to closest pair in higher dimensions is optimal up to an n^o(1) factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.
Cite as
Scott Aaronson, Nai-Hui Chia, Han-Hsuan Lin, Chunhao Wang, and Ruizhe Zhang. On the Quantum Complexity of Closest Pair and Related Problems. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 16:1-16:43, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{aaronson_et_al:LIPIcs.CCC.2020.16,
author = {Aaronson, Scott and Chia, Nai-Hui and Lin, Han-Hsuan and Wang, Chunhao and Zhang, Ruizhe},
title = {{On the Quantum Complexity of Closest Pair and Related Problems}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {16:1--16:43},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.16},
URN = {urn:nbn:de:0030-drops-125681},
doi = {10.4230/LIPIcs.CCC.2020.16},
annote = {Keywords: Closest pair, Quantum computing, Quantum fine grained reduction, Quantum strong exponential time hypothesis, Fine grained complexity}
}
Document
Limits of Preprocessing
Abstract
It is a classical result that the inner product function cannot be computed by an AC⁰ circuit [Merrick L. Furst et al., 1981; Miklós Ajtai, 1983; Johan Håstad, 1986]. It is conjectured that this holds even if we allow arbitrary preprocessing of each of the two inputs separately. We prove this conjecture when the preprocessing of one of the inputs is limited to output n + n/(log^{ω(1)} n) bits. Our methods extend to many other functions, including pseudorandom functions, and imply a (weak but nontrivial) limitation on the power of encoding inputs in low-complexity cryptography. Finally, under cryptographic assumptions, we relate the question of proving variants of the main conjecture with the question of learning AC⁰ under simple input distributions.
Cite as
Yuval Filmus, Yuval Ishai, Avi Kaplan, and Guy Kindler. Limits of Preprocessing. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 17:1-17:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{filmus_et_al:LIPIcs.CCC.2020.17,
author = {Filmus, Yuval and Ishai, Yuval and Kaplan, Avi and Kindler, Guy},
title = {{Limits of Preprocessing}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {17:1--17:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.17},
URN = {urn:nbn:de:0030-drops-125697},
doi = {10.4230/LIPIcs.CCC.2020.17},
annote = {Keywords: circuit, communication complexity, IPPP, preprocessing, PRF, simultaneous messages}
}
Document
Sign Rank vs Discrepancy
Abstract
Sign-rank and discrepancy are two central notions in communication complexity. The seminal work of Babai, Frankl, and Simon from 1986 initiated an active line of research that investigates the gap between these two notions. In this article, we establish the strongest possible separation by constructing a boolean matrix whose sign-rank is only 3, and yet its discrepancy is 2^{-Ω(n)}. We note that every matrix of sign-rank 2 has discrepancy n^{-O(1)}.
Our result in particular implies that there are boolean functions with O(1) unbounded error randomized communication complexity while having Ω(n) weakly unbounded error randomized communication complexity.
Cite as
Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Sign Rank vs Discrepancy. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{hatami_et_al:LIPIcs.CCC.2020.18,
author = {Hatami, Hamed and Hosseini, Kaave and Lovett, Shachar},
title = {{Sign Rank vs Discrepancy}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {18:1--18:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.18},
URN = {urn:nbn:de:0030-drops-125700},
doi = {10.4230/LIPIcs.CCC.2020.18},
annote = {Keywords: Discrepancy, sign rank, Unbounded-error communication complexity, weakly unbounded error communication complexity}
}
Document
On the Complexity of Modulo-q Arguments and the Chevalley - Warning Theorem
Abstract
We study the search problem class PPA_q defined as a modulo-q analog of the well-known polynomial parity argument class PPA introduced by Papadimitriou (JCSS 1994). Our first result shows that this class can be characterized in terms of PPA_p for prime p.
Our main result is to establish that an explicit version of a search problem associated to the Chevalley - Warning theorem is complete for PPA_p for prime p. This problem is natural in that it does not explicitly involve circuits as part of the input. It is the first such complete problem for PPA_p when p ≥ 3.
Finally we discuss connections between Chevalley-Warning theorem and the well-studied short integer solution problem and survey the structural properties of PPA_q.
Cite as
Mika Göös, Pritish Kamath, Katerina Sotiraki, and Manolis Zampetakis. On the Complexity of Modulo-q Arguments and the Chevalley - Warning Theorem. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 19:1-19:42, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{goos_et_al:LIPIcs.CCC.2020.19,
author = {G\"{o}\"{o}s, Mika and Kamath, Pritish and Sotiraki, Katerina and Zampetakis, Manolis},
title = {{On the Complexity of Modulo-q Arguments and the Chevalley - Warning Theorem}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {19:1--19:42},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.19},
URN = {urn:nbn:de:0030-drops-125712},
doi = {10.4230/LIPIcs.CCC.2020.19},
annote = {Keywords: Total NP Search Problems, Modulo-q arguments, Chevalley - Warning Theorem}
}
Document
Non-Disjoint Promise Problems from Meta-Computational View of Pseudorandom Generator Constructions
Abstract
The standard notion of promise problem is a pair of disjoint sets of instances, each of which is regarded as Yes and No instances, respectively, and the task of solving a promise problem is to distinguish these two sets of instances. In this paper, we introduce a set of new promise problems which are conjectured to be non-disjoint, and prove that hardness of these "non-disjoint" promise problems gives rise to the existence of hitting set generators (and vice versa). We do this by presenting a general principle which converts any black-box construction of a pseudorandom generator into the existence of a hitting set generator whose security is based on hardness of some "non-disjoint" promise problem (via a non-black-box security reduction).
Applying the principle to cryptographic pseudorandom generators, we introduce
- The Gap(K^SAT vs K) Problem: Given a string x and a parameter s, distinguish whether the polynomial-time-bounded SAT-oracle Kolmogorov complexity of x is at most s, or the polynomial-time-bounded Kolmogorov complexity of x (without SAT oracle) is at least s + O(log|x|).
If Gap(K^SAT vs K) is NP-hard, then the worst-case and average-case complexity of PH is equivalent. Under the plausible assumption that E^NP ≠ E, the promise problem is non-disjoint. These results generalize the non-black-box worst-case to average-case reductions of Hirahara [Hirahara, 2018] and improve the approximation error from Õ(√n) to O(log n).
Applying the principle to complexity-theoretic pseudorandom generators, we introduce a family of Meta-computational Circuit Lower-bound Problems (MCLPs), which are problems of distinguishing the truth tables of explicit functions from hard functions. Our results generalize the hardness versus randomness framework and identify problems whose circuit lower bounds characterize the existence of hitting set generators. For example, we introduce
- The E vs SIZE(2^o(n)) Problem: Given the truth table of a function f, distinguish whether f is computable in exponential time or requires exponential-size circuits to compute.
A nearly-linear AC⁰ ∘ XOR circuit size lower bound for this promise problem is equivalent to the existence of a logarithmic-seed-length hitting set generator for AC⁰ ∘ XOR. Under the plausible assumption that E ⊈ SIZE(2^o(n)), the promise problem is non-disjoint (and thus the minimum circuit size is infinity). This is the first result that provides the exact characterization of the existence of a hitting set generator secure against ℭ by the worst-case lower bound against ℭ for a circuit class ℭ = AC⁰ ∘ XOR ⊂ TC⁰. In addition, we prove that a nearly-linear size lower bound against co-nondeterministic read-once branching programs for some "non-disjoint" promise problem is sufficient for resolving RL = L.
We also establish the equivalence between the existence of a derandomization algorithm for uniform algorithms and a uniform lower bound for a problem of approximating Levin’s Kt-complexity.
Cite as
Shuichi Hirahara. Non-Disjoint Promise Problems from Meta-Computational View of Pseudorandom Generator Constructions. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 20:1-20:47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{hirahara:LIPIcs.CCC.2020.20,
author = {Hirahara, Shuichi},
title = {{Non-Disjoint Promise Problems from Meta-Computational View of Pseudorandom Generator Constructions}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {20:1--20:47},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.20},
URN = {urn:nbn:de:0030-drops-125720},
doi = {10.4230/LIPIcs.CCC.2020.20},
annote = {Keywords: meta-complexity, pseudorandom generator, hitting set generator}
}
Document
Algebraic Branching Programs, Border Complexity, and Tangent Spaces
Abstract
Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices. This means that the set of noncommutative polynomials with ABP width complexity at most k is Zariski-closed, an important property in geometric complexity theory. It follows that approximations cannot help to reduce the required ABP width.
It was mentioned by Forbes that this result would probably break when going from single-(source,sink) ABPs to trace ABPs. We prove that this is correct. Moreover, we study the commutative monotone setting and prove a result similar to Nisan, but concerning the analytic closure. We observe the same behavior here: The set of polynomials with ABP width complexity at most k is closed for single-(source,sink) ABPs and not closed for trace ABPs. The proofs reveal an intriguing connection between tangent spaces and the vector space of flows on the ABP. We close with additional observations on VQP and the closure of VNP which allows us to establish a separation between the two classes.
Cite as
Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, and Nitin Saurabh. Algebraic Branching Programs, Border Complexity, and Tangent Spaces. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{blaser_et_al:LIPIcs.CCC.2020.21,
author = {Bl\"{a}ser, Markus and Ikenmeyer, Christian and Mahajan, Meena and Pandey, Anurag and Saurabh, Nitin},
title = {{Algebraic Branching Programs, Border Complexity, and Tangent Spaces}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {21:1--21:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.21},
URN = {urn:nbn:de:0030-drops-125733},
doi = {10.4230/LIPIcs.CCC.2020.21},
annote = {Keywords: Algebraic Branching Programs, Border Complexity, Tangent Spaces, Lower Bounds, Geometric Complexity Theory, Flows, VQP, VNP}
}
Document
NP-Hardness of Circuit Minimization for Multi-Output Functions
Abstract
Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive.
In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n → {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators.
Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless 𝖯 = 𝖭𝖯, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions.
Cite as
Rahul Ilango, Bruno Loff, and Igor C. Oliveira. NP-Hardness of Circuit Minimization for Multi-Output Functions. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 22:1-22:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{ilango_et_al:LIPIcs.CCC.2020.22,
author = {Ilango, Rahul and Loff, Bruno and Oliveira, Igor C.},
title = {{NP-Hardness of Circuit Minimization for Multi-Output Functions}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {22:1--22:36},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.22},
URN = {urn:nbn:de:0030-drops-125744},
doi = {10.4230/LIPIcs.CCC.2020.22},
annote = {Keywords: MCSP, circuit minimization, communication complexity, Boolean circuit}
}
Document
A Super-Quadratic Lower Bound for Depth Four Arithmetic Circuits
Abstract
We show an Ω̃(n^2.5) lower bound for general depth four arithmetic circuits computing an explicit n-variate degree-Θ(n) multilinear polynomial over any field of characteristic zero. To our knowledge, and as stated in the survey [Amir Shpilka and Amir Yehudayoff, 2010], no super-quadratic lower bound was known for depth four circuits over fields of characteristic ≠ 2 before this work. The previous best lower bound is Ω̃(n^1.5) [Abhijat Sharma, 2017], which is a slight quantitative improvement over the roughly Ω(n^1.33) bound obtained by invoking the super-linear lower bound for constant depth circuits in [Ran Raz, 2010; Victor Shoup and Roman Smolensky, 1997].
Our lower bound proof follows the approach of the almost cubic lower bound for depth three circuits in [Neeraj Kayal et al., 2016] by replacing the shifted partials measure with a suitable variant of the projected shifted partials measure, but it differs from [Neeraj Kayal et al., 2016]’s proof at a crucial step - namely, the way "heavy" product gates are handled. Loosely speaking, a heavy product gate has a relatively high fan-in. Product gates of a depth three circuit compute products of affine forms, and so, it is easy to prune Θ(n) many heavy product gates by projecting the circuit to a low-dimensional affine subspace [Neeraj Kayal et al., 2016; Amir Shpilka and Avi Wigderson, 2001]. However, in a depth four circuit, the second (from the top) layer of product gates compute products of polynomials having arbitrary degree, and hence it was not clear how to prune such heavy product gates from the circuit. We show that heavy product gates can also be eliminated from a depth four circuit by projecting the circuit to a low-dimensional affine subspace, unless the heavy gates together account for Ω̃(n^2.5) size. This part of our argument is inspired by a well-known greedy approximation algorithm for the weighted set-cover problem.
Cite as
Nikhil Gupta, Chandan Saha, and Bhargav Thankey. A Super-Quadratic Lower Bound for Depth Four Arithmetic Circuits. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 23:1-23:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{gupta_et_al:LIPIcs.CCC.2020.23,
author = {Gupta, Nikhil and Saha, Chandan and Thankey, Bhargav},
title = {{A Super-Quadratic Lower Bound for Depth Four Arithmetic Circuits}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {23:1--23:31},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.23},
URN = {urn:nbn:de:0030-drops-125757},
doi = {10.4230/LIPIcs.CCC.2020.23},
annote = {Keywords: depth four arithmetic circuits, Projected Shifted Partials, super-quadratic lower bound}
}
Document
Multiparty Karchmer - Wigderson Games and Threshold Circuits
Abstract
We suggest a generalization of Karchmer - Wigderson communication games to the multiparty setting. Our generalization turns out to be tightly connected to circuits consisting of threshold gates. This allows us to obtain new explicit constructions of such circuits for several functions. In particular, we provide an explicit (polynomial-time computable) log-depth monotone formula for Majority function, consisting only of 3-bit majority gates and variables. This resolves a conjecture of Cohen et al. (CRYPTO 2013).
Cite as
Alexander Kozachinskiy and Vladimir Podolskii. Multiparty Karchmer - Wigderson Games and Threshold Circuits. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 24:1-24:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kozachinskiy_et_al:LIPIcs.CCC.2020.24,
author = {Kozachinskiy, Alexander and Podolskii, Vladimir},
title = {{Multiparty Karchmer - Wigderson Games and Threshold Circuits}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {24:1--24:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.24},
URN = {urn:nbn:de:0030-drops-125767},
doi = {10.4230/LIPIcs.CCC.2020.24},
annote = {Keywords: Karchmer-Wigderson Games, Threshold Circuits, threshold gates, majority function}
}
Document
Optimal Error Pseudodistributions for Read-Once Branching Programs
Abstract
In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom generator for length n and width w read-once branching programs with seed length O(log n⋅ log(nw)+log n⋅log(1/ε)) and error ε. It remains a central question to reduce the seed length to O(log (nw/ε)), which would prove that 𝐁𝐏𝐋 = 𝐋. However, there has been no improvement on Nisan’s construction for the case n = w, which is most relevant to space-bounded derandomization.
Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced the notion of a pseudorandom pseudo-distribution (PRPD) and gave an explicit construction of a PRPD with seed length Õ(log n⋅ log(nw)+log(1/ε)). A PRPD is a relaxation of a pseudorandom generator, which suffices for derandomizing 𝐁𝐏𝐋 and also implies a hitting set. Unfortunately, their construction is quite involved and complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler hitting set generator with seed length O(log n⋅ log(nw)+log(1/ε)), but their techniques are restricted to hitting sets.
In this work, we construct a PRPD with seed length O(log n⋅ log (nw)⋅ log log(nw)+log(1/ε)). This improves upon the construction by Braverman, Cogen and Garg by a O(log log(1/ε)) factor, and is optimal in the small error regime. In addition, we believe our construction and analysis to be simpler than the work of Braverman, Cohen and Garg.
Cite as
Eshan Chattopadhyay and Jyun-Jie Liao. Optimal Error Pseudodistributions for Read-Once Branching Programs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 25:1-25:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2020.25,
author = {Chattopadhyay, Eshan and Liao, Jyun-Jie},
title = {{Optimal Error Pseudodistributions for Read-Once Branching Programs}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {25:1--25:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.25},
URN = {urn:nbn:de:0030-drops-125779},
doi = {10.4230/LIPIcs.CCC.2020.25},
annote = {Keywords: Derandomization, explicit constructions, space-bounded computation}
}
Document
Circuit Lower Bounds from NP-Hardness of MCSP Under Turing Reductions
Abstract
The fundamental Minimum Circuit Size Problem is a well-known example of a problem that is neither known to be in 𝖯 nor known to be NP-hard. Kabanets and Cai [Kabanets and Cai, 2000] showed that if MCSP is NP-hard under "natural" m-reductions, superpolynomial circuit lower bounds for exponential time would follow. This has triggered a long line of work on understanding the power of reductions to MCSP.
Nothing was known so far about consequences of NP-hardness of MCSP under general Turing reductions. In this work, we consider two structured kinds of Turing reductions: parametric honest reductions and natural reductions. The latter generalize the natural reductions of Kabanets and Cai to the case of Turing-reductions. We show that NP-hardness of MCSP under these kinds of Turing-reductions imply superpolynomial circuit lower bounds for exponential time.
Cite as
Michael Saks and Rahul Santhanam. Circuit Lower Bounds from NP-Hardness of MCSP Under Turing Reductions. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 26:1-26:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{saks_et_al:LIPIcs.CCC.2020.26,
author = {Saks, Michael and Santhanam, Rahul},
title = {{Circuit Lower Bounds from NP-Hardness of MCSP Under Turing Reductions}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {26:1--26:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.26},
URN = {urn:nbn:de:0030-drops-125786},
doi = {10.4230/LIPIcs.CCC.2020.26},
annote = {Keywords: Minimum Circuit Size Problem, Turing reductions, circuit lower bounds}
}
Document
Finding Small Satisfying Assignments Faster Than Brute Force: A Fine-Grained Perspective into Boolean Constraint Satisfaction
Abstract
To study the question under which circumstances small solutions can be found faster than by exhaustive search (and by how much), we study the fine-grained complexity of Boolean constraint satisfaction with size constraint exactly k. More precisely, we aim to determine, for any finite constraint family, the optimal running time f(k)n^g(k) required to find satisfying assignments that set precisely k of the n variables to 1.
Under central hardness assumptions on detecting cliques in graphs and 3-uniform hypergraphs, we give an almost tight characterization of g(k) into four regimes:
1) Brute force is essentially best-possible, i.e., g(k) = (1 ± o(1))k,
2) the best algorithms are as fast as current k-clique algorithms, i.e., g(k) = (ω/3 ± o(1))k,
3) the exponent has sublinear dependence on k with g(k) ∈ [Ω(∛k), O(√k)], or
4) the problem is fixed-parameter tractable, i.e., g(k) = O(1).
This yields a more fine-grained perspective than a previous FPT/W[1]-hardness dichotomy (Marx, Computational Complexity 2005). Our most interesting technical contribution is a f(k)n^(4√k)-time algorithm for SubsetSum with precedence constraints parameterized by the target k - particularly the approach, based on generalizing a bound on the Frobenius coin problem to a setting with precedence constraints, might be of independent interest.
Cite as
Marvin Künnemann and Dániel Marx. Finding Small Satisfying Assignments Faster Than Brute Force: A Fine-Grained Perspective into Boolean Constraint Satisfaction. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 27:1-27:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kunnemann_et_al:LIPIcs.CCC.2020.27,
author = {K\"{u}nnemann, Marvin and Marx, D\'{a}niel},
title = {{Finding Small Satisfying Assignments Faster Than Brute Force: A Fine-Grained Perspective into Boolean Constraint Satisfaction}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {27:1--27:28},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.27},
URN = {urn:nbn:de:0030-drops-125791},
doi = {10.4230/LIPIcs.CCC.2020.27},
annote = {Keywords: Fine-grained complexity theory, algorithmic classification theorem, multivariate algorithms and complexity, constraint satisfaction problems, satisfiability}
}
Document
Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs
Abstract
We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense graphs as in [Raz '04] and [Razborov '03, '04]. We obtain our results by revisiting Razborov’s pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.
Cite as
Susanna F. de Rezende, Jakob Nordström, Kilian Risse, and Dmitry Sokolov. Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 28:1-28:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{derezende_et_al:LIPIcs.CCC.2020.28,
author = {de Rezende, Susanna F. and Nordstr\"{o}m, Jakob and Risse, Kilian and Sokolov, Dmitry},
title = {{Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {28:1--28:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.28},
URN = {urn:nbn:de:0030-drops-125804},
doi = {10.4230/LIPIcs.CCC.2020.28},
annote = {Keywords: proof complexity, resolution, weak pigeonhole principle, perfect matching, sparse graphs}
}
Document
Groups with ALOGTIME-Hard Word Problems and PSPACE-Complete Circuit Value Problems
Abstract
We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk’s group and Thompson’s groups, we prove that their word problem is ALOGTIME-hard. For some of these groups (including Grigorchuk’s group and Thompson’s groups) we prove that the circuit value problem (which is equivalent to the circuit evaluation problem) is PSPACE-complete.
Cite as
Laurent Bartholdi, Michael Figelius, Markus Lohrey, and Armin Weiß. Groups with ALOGTIME-Hard Word Problems and PSPACE-Complete Circuit Value Problems. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 29:1-29:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bartholdi_et_al:LIPIcs.CCC.2020.29,
author = {Bartholdi, Laurent and Figelius, Michael and Lohrey, Markus and Wei{\ss}, Armin},
title = {{Groups with ALOGTIME-Hard Word Problems and PSPACE-Complete Circuit Value Problems}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {29:1--29:29},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.29},
URN = {urn:nbn:de:0030-drops-125814},
doi = {10.4230/LIPIcs.CCC.2020.29},
annote = {Keywords: NC^1-hardness, word problem, G-programs, straight-line programs, non-solvable groups, self-similar groups, Thompson’s groups, Grigorchuk’s group}
}
Document
Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams
Abstract
In this paper, we give simple optimal lower bounds on the one-way two-party communication complexity of approximate Maximum Matching and Minimum Vertex Cover with deletions. In our model, Alice holds a set of edges and sends a single message to Bob. Bob holds a set of edge deletions, which form a subset of Alice’s edges, and needs to report a large matching or a small vertex cover in the graph spanned by the edges that are not deleted. Our results imply optimal space lower bounds for insertion-deletion streaming algorithms for Maximum Matching and Minimum Vertex Cover.
Previously, Assadi et al. [SODA 2016] gave an optimal space lower bound for insertion-deletion streaming algorithms for Maximum Matching via the simultaneous model of communication. Our lower bound is simpler and stronger in several aspects: The lower bound of Assadi et al. only holds for algorithms that (1) are able to process streams that contain a triple exponential number of deletions in n, the number of vertices of the input graph; (2) are able to process multi-graphs; and (3) never output edges that do not exist in the input graph when the randomized algorithm errs. In contrast, our lower bound even holds for algorithms that (1) rely on short (O(n²)-length) input streams; (2) are only able to process simple graphs; and (3) may output non-existing edges when the algorithm errs.
Cite as
Jacques Dark and Christian Konrad. Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{dark_et_al:LIPIcs.CCC.2020.30,
author = {Dark, Jacques and Konrad, Christian},
title = {{Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {30:1--30:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.30},
URN = {urn:nbn:de:0030-drops-125824},
doi = {10.4230/LIPIcs.CCC.2020.30},
annote = {Keywords: Maximum matching, Minimum vertex cover, Dynamic graph streams, Communication complexity, Lower bounds}
}
Document
Connecting Perebor Conjectures: Towards a Search to Decision Reduction for Minimizing Formulas
Abstract
A longstanding open question is whether there is an equivalence between the computational task of determining the minimum size of any circuit computing a given function and the task of producing a minimum-sized circuit for a given function. While it is widely conjectured that both tasks require "perebor," or brute-force search, researchers have not yet ruled out the possibility that the search problem requires exponential time but the decision problem has a linear time algorithm.
In this paper, we make progress in connecting the search and decision complexity of minimizing formulas. Let MFSP denote the problem that takes as input the truth table of a Boolean function f and an integer size parameter s and decides whether there is a formula for f of size at most s. Let Search- denote the corresponding search problem where one has to output some optimal formula for computing f.
Our main result is that given an oracle to MFSP, one can solve Search-MFSP in time polynomial in the length N of the truth table of f and the number t of "near-optimal" formulas for f, in particular O(N⁶t²)-time. While the quantity t is not well understood, we use this result (and some extensions) to prove that given an oracle to MFSP:
- there is a deterministic 2^O(N/(log log N))-time oracle algorithm for solving Search-MFSP on all but a o(1)-fraction of instances, and
- there is a randomized O(2^.67N)-time oracle algorithm for solving Search-MFSP on all instances. Intriguingly, the main idea behind our algorithms is in some sense a "reverse application" of the gate elimination technique.
Cite as
Rahul Ilango. Connecting Perebor Conjectures: Towards a Search to Decision Reduction for Minimizing Formulas. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 31:1-31:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{ilango:LIPIcs.CCC.2020.31,
author = {Ilango, Rahul},
title = {{Connecting Perebor Conjectures: Towards a Search to Decision Reduction for Minimizing Formulas}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {31:1--31:35},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.31},
URN = {urn:nbn:de:0030-drops-125834},
doi = {10.4230/LIPIcs.CCC.2020.31},
annote = {Keywords: minimum circuit size problem, minimum formula size problem, gate elimination, search to decision reduction, self-reducibility}
}
Document
Quantum Query-To-Communication Simulation Needs a Logarithmic Overhead
Abstract
Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function f:{-1,1}ⁿ → {-1,1} and •:{-1,1}² → {-1,1} the two-party bounded-error quantum communication complexity of (f ∘ •) is O(Q(f) log n), where Q(f) is the bounded-error quantum query complexity of f. Note that the bounded-error randomized communication complexity of (f ∘ •) is bounded by O(R(f)), where R(f) denotes the bounded-error randomized query complexity of f. Thus, the BCW simulation has an extra O(log n) factor appearing that is absent in classical simulation. A natural question is if this factor can be avoided. Razborov (IZV MATH'03) showed that the bounded-error quantum communication complexity of Set-Disjointness is Ω(√n). The BCW simulation yields an upper bound of O(√n log n). Høyer and de Wolf (STACS'02) showed that this can be reduced to c^(log^* n) for some constant c, and subsequently Aaronson and Ambainis (FOCS'03) showed that this factor can be made a constant. That is, the quantum communication complexity of the Set-Disjointness function (which is NOR_n ∘ ∧) is O(Q(NOR_n)).
Perhaps somewhat surprisingly, we show that when • = ⊕, then the extra log n factor in the BCW simulation is unavoidable. In other words, we exhibit a total function F:{-1,1}ⁿ → {-1,1} such that Q^{cc}(F ∘ ⊕) = Θ(Q(F) log n).
To the best of our knowledge, it was not even known prior to this work whether there existed a total function F and 2-bit function •, such that Q^{cc}(F ∘ •) = ω(Q(F)).
Cite as
Sourav Chakraborty, Arkadev Chattopadhyay, Nikhil S. Mande, and Manaswi Paraashar. Quantum Query-To-Communication Simulation Needs a Logarithmic Overhead. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 32:1-32:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{chakraborty_et_al:LIPIcs.CCC.2020.32,
author = {Chakraborty, Sourav and Chattopadhyay, Arkadev and Mande, Nikhil S. and Paraashar, Manaswi},
title = {{Quantum Query-To-Communication Simulation Needs a Logarithmic Overhead}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {32:1--32:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.32},
URN = {urn:nbn:de:0030-drops-125842},
doi = {10.4230/LIPIcs.CCC.2020.32},
annote = {Keywords: Quantum query complexity, quantum communication complexity, approximate degree, approximate spectral norm}
}
Document
Factorization of Polynomials Given By Arithmetic Branching Programs
Abstract
Given a multivariate polynomial computed by an arithmetic branching program (ABP) of size s, we show that all its factors can be computed by arithmetic branching programs of size poly(s). Kaltofen gave a similar result for polynomials computed by arithmetic circuits. The previously known best upper bound for ABP-factors was poly(s^(log s)).
Cite as
Amit Sinhababu and Thomas Thierauf. Factorization of Polynomials Given By Arithmetic Branching Programs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{sinhababu_et_al:LIPIcs.CCC.2020.33,
author = {Sinhababu, Amit and Thierauf, Thomas},
title = {{Factorization of Polynomials Given By Arithmetic Branching Programs}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {33:1--33:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.33},
URN = {urn:nbn:de:0030-drops-125854},
doi = {10.4230/LIPIcs.CCC.2020.33},
annote = {Keywords: Arithmetic Branching Program, Multivariate Polynomial Factorization, Hensel Lifting, Newton Iteration, Hardness vs Randomness}
}
Document
On the Complexity of Branching Proofs
Abstract
We consider the task of proving integer infeasibility of a bounded convex K in ℝⁿ using a general branching proof system. In a general branching proof, one constructs a branching tree by adding an integer disjunction 𝐚𝐱 ≤ b or 𝐚𝐱 ≥ b+1, 𝐚 ∈ ℤⁿ, b ∈ ℤ, at each node, such that the leaves of the tree correspond to empty sets (i.e., K together with the inequalities picked up from the root to leaf is empty).
Recently, Beame et al (ITCS 2018), asked whether the bit size of the coefficients in a branching proof, which they named stabbing planes (SP) refutations, for the case of polytopes derived from SAT formulas, can be assumed to be polynomial in n. We resolve this question in the affirmative, by showing that any branching proof can be recompiled so that the normals of the disjunctions have coefficients of size at most (n R)^O(n²), where R ∈ ℕ is the radius of an 𝓁₁ ball containing K, while increasing the number of nodes in the branching tree by at most a factor O(n). Our recompilation techniques works by first replacing each disjunction using an iterated Diophantine approximation, introduced by Frank and Tardos (Combinatorica 1986), and proceeds by "fixing up" the leaves of the tree using judiciously added Chvátal-Gomory (CG) cuts.
As our second contribution, we show that Tseitin formulas, an important class of infeasible SAT instances, have quasi-polynomial sized cutting plane (CP) refutations. This disproves a conjecture that Tseitin formulas are (exponentially) hard for CP. Our upper bound follows by recompiling the quasi-polynomial sized SP refutations for Tseitin formulas due to Beame et al, which have a special enumerative form, into a CP proof of the same length using a serialization technique of Cook et al (Discrete Appl. Math. 1987).
As our final contribution, we give a simple family of polytopes in [0,1]ⁿ requiring exponential sized branching proofs.
Cite as
Daniel Dadush and Samarth Tiwari. On the Complexity of Branching Proofs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 34:1-34:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{dadush_et_al:LIPIcs.CCC.2020.34,
author = {Dadush, Daniel and Tiwari, Samarth},
title = {{On the Complexity of Branching Proofs}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {34:1--34:35},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.34},
URN = {urn:nbn:de:0030-drops-125863},
doi = {10.4230/LIPIcs.CCC.2020.34},
annote = {Keywords: Branching Proofs, Cutting Planes, Diophantine Approximation, Integer Programming, Stabbing Planes, Tseitin Formulas}
}
Document
Geometric Rank of Tensors and Subrank of Matrix Multiplication
Abstract
Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and hypergraphs. We prove that the geometric rank is an upper bound on the subrank of tensors and the independence number of hypergraphs. We prove that the geometric rank is smaller than the slice rank of Tao, and relate geometric rank to the analytic rank of Gowers and Wolf in an asymptotic fashion. As a first application, we use geometric rank to prove a tight upper bound on the (border) subrank of the matrix multiplication tensors, matching Strassen’s well-known lower bound from 1987.
Cite as
Swastik Kopparty, Guy Moshkovitz, and Jeroen Zuiddam. Geometric Rank of Tensors and Subrank of Matrix Multiplication. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 35:1-35:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{kopparty_et_al:LIPIcs.CCC.2020.35,
author = {Kopparty, Swastik and Moshkovitz, Guy and Zuiddam, Jeroen},
title = {{Geometric Rank of Tensors and Subrank of Matrix Multiplication}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {35:1--35:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.35},
URN = {urn:nbn:de:0030-drops-125874},
doi = {10.4230/LIPIcs.CCC.2020.35},
annote = {Keywords: Algebraic complexity theory, Extremal combinatorics, Tensors, Bias, Analytic rank, Algebraic geometry, Matrix multiplication}
}
Document
Hardness of Bounded Distance Decoding on Lattices in 𝓁_p Norms
Abstract
Bounded Distance Decoding BDD_{p,α} is the problem of decoding a lattice when the target point is promised to be within an α factor of the minimum distance of the lattice, in the 𝓁_p norm. We prove that BDD_{p, α} is NP-hard under randomized reductions where α → 1/2 as p → ∞ (and for α = 1/2 when p = ∞), thereby showing the hardness of decoding for distances approaching the unique-decoding radius for large p. We also show fine-grained hardness for BDD_{p,α}. For example, we prove that for all p ∈ [1,∞) ⧵ 2ℤ and constants C > 1, ε > 0, there is no 2^((1-ε)n/C)-time algorithm for BDD_{p,α} for some constant α (which approaches 1/2 as p → ∞), assuming the randomized Strong Exponential Time Hypothesis (SETH). Moreover, essentially all of our results also hold (under analogous non-uniform assumptions) for BDD with preprocessing, in which unbounded precomputation can be applied to the lattice before the target is available.
Compared to prior work on the hardness of BDD_{p,α} by Liu, Lyubashevsky, and Micciancio (APPROX-RANDOM 2008), our results improve the values of α for which the problem is known to be NP-hard for all p > p₁ ≈ 4.2773, and give the very first fine-grained hardness for BDD (in any norm). Our reductions rely on a special family of "locally dense" lattices in 𝓁_p norms, which we construct by modifying the integer-lattice sparsification technique of Aggarwal and Stephens-Davidowitz (STOC 2018).
Cite as
Huck Bennett and Chris Peikert. Hardness of Bounded Distance Decoding on Lattices in 𝓁_p Norms. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bennett_et_al:LIPIcs.CCC.2020.36,
author = {Bennett, Huck and Peikert, Chris},
title = {{Hardness of Bounded Distance Decoding on Lattices in 𝓁\underlinep Norms}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {36:1--36:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.36},
URN = {urn:nbn:de:0030-drops-125881},
doi = {10.4230/LIPIcs.CCC.2020.36},
annote = {Keywords: Lattices, Bounded Distance Decoding, NP-hardness, Fine-Grained Complexity}
}
Document
Algebraic Hardness Versus Randomness in Low Characteristic
Abstract
We show that lower bounds for explicit constant-variate polynomials over fields of characteristic p > 0 are sufficient to derandomize polynomial identity testing over fields of characteristic p. In this setting, existing work on hardness-randomness tradeoffs for polynomial identity testing requires either the characteristic to be sufficiently large or the notion of hardness to be stronger than the standard syntactic notion of hardness used in algebraic complexity. Our results make no restriction on the characteristic of the field and use standard notions of hardness.
We do this by combining the Kabanets-Impagliazzo generator with a white-box procedure to take p-th roots of circuits computing a p-th power over fields of characteristic p. When the number of variables appearing in the circuit is bounded by some constant, this procedure turns out to be efficient, which allows us to bypass difficulties related to factoring circuits in characteristic p.
We also combine the Kabanets-Impagliazzo generator with recent "bootstrapping" results in polynomial identity testing to show that a sufficiently-hard family of explicit constant-variate polynomials yields a near-complete derandomization of polynomial identity testing. This result holds over fields of both zero and positive characteristic and complements a recent work of Guo, Kumar, Saptharishi, and Solomon, who obtained a slightly stronger statement over fields of characteristic zero.
Cite as
Robert Andrews. Algebraic Hardness Versus Randomness in Low Characteristic. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 37:1-37:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{andrews:LIPIcs.CCC.2020.37,
author = {Andrews, Robert},
title = {{Algebraic Hardness Versus Randomness in Low Characteristic}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {37:1--37:32},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.37},
URN = {urn:nbn:de:0030-drops-125895},
doi = {10.4230/LIPIcs.CCC.2020.37},
annote = {Keywords: Polynomial identity testing, hardness versus randomness, low characteristic}
}
Document
Sum of Squares Bounds for the Ordering Principle
Abstract
In this paper, we analyze the sum of squares hierarchy (SOS) on the ordering principle on n elements (which has N = Θ(n²) variables). We prove that degree O(√nlog(n)) SOS can prove the ordering principle. We then show that this upper bound is essentially tight by proving that for any ε > 0, SOS requires degree Ω(n^(1/2 - ε)) to prove the ordering principle.
Cite as
Aaron Potechin. Sum of Squares Bounds for the Ordering Principle. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 38:1-38:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{potechin:LIPIcs.CCC.2020.38,
author = {Potechin, Aaron},
title = {{Sum of Squares Bounds for the Ordering Principle}},
booktitle = {35th Computational Complexity Conference (CCC 2020)},
pages = {38:1--38:37},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-156-6},
ISSN = {1868-8969},
year = {2020},
volume = {169},
editor = {Saraf, Shubhangi},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.38},
URN = {urn:nbn:de:0030-drops-125900},
doi = {10.4230/LIPIcs.CCC.2020.38},
annote = {Keywords: sum of squares hierarchy, proof complexity, ordering principle}
}