Volume
LIPIcs, Volume 234
37th Computational Complexity Conference (CCC 2022)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Computational Complexity Conference (CCC)
Event
CCC 2022, July 20-23, 2022, Philadelphia, PA, USAEditor
Publication Details
- published at: 2022-07-11
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-241-9
- DBLP: db/conf/coco/coco2022
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Document
Complete Volume
LIPIcs, Volume 234, CCC 2022, Complete Volume
Abstract
LIPIcs, Volume 234, CCC 2022, Complete Volume
Cite as
37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 1-960, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@Proceedings{lovett:LIPIcs.CCC.2022,
title = {{LIPIcs, Volume 234, CCC 2022, Complete Volume}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {1--960},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022},
URN = {urn:nbn:de:0030-drops-165614},
doi = {10.4230/LIPIcs.CCC.2022},
annote = {Keywords: LIPIcs, Volume 234, CCC 2022, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{lovett:LIPIcs.CCC.2022.0,
author = {Lovett, Shachar},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {0:i--0:xvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.0},
URN = {urn:nbn:de:0030-drops-165621},
doi = {10.4230/LIPIcs.CCC.2022.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
The Approximate Degree of Bipartite Perfect Matching
Abstract
The approximate degree of a Boolean function is the least degree of a real multilinear polynomial approximating it in the 𝓁_∞-norm over the Boolean hypercube. We show that the approximate degree of the Bipartite Perfect Matching function, which is the indicator over all bipartite graphs having a perfect matching of order n, is Θ̃(n^(3/2)).
The upper bound is obtained by fully characterizing the unique multilinear polynomial representing the Boolean dual of the perfect matching function, over the reals. Crucially, we show that this polynomial has very small 𝓁₁-norm - only exponential in Θ(n log n). The lower bound follows by bounding the spectral sensitivity of the perfect matching function, which is the spectral radius of its cut-graph on the hypercube [Aaronson et al., 2021; Huang, 2019]. We show that the spectral sensitivity of perfect matching is exactly Θ(n^(3/2)).
Cite as
Gal Beniamini. The Approximate Degree of Bipartite Perfect Matching. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 1:1-1:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{beniamini:LIPIcs.CCC.2022.1,
author = {Beniamini, Gal},
title = {{The Approximate Degree of Bipartite Perfect Matching}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {1:1--1:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.1},
URN = {urn:nbn:de:0030-drops-165634},
doi = {10.4230/LIPIcs.CCC.2022.1},
annote = {Keywords: Bipartite Perfect Matching, Boolean Functions, Approximate Degree}
}
Document
On the Satisfaction Probability of k-CNF Formulas
Abstract
The satisfaction probability σ(ϕ) := Pr_{β:vars(ϕ) → {0,1}}[β ⊧ ϕ] of a propositional formula ϕ is the likelihood that a random assignment β makes the formula true. We study the complexity of the problem kSAT-PROB_{> δ} = {ϕ is a kCNF formula ∣ σ(ϕ) > δ} for fixed k and δ. While 3SAT-PROB_{> 0} = 3SAT is NP-complete and SAT-PROB}_{> 1/2} is PP-complete, Akmal and Williams recently showed 3SAT-PROB_{> 1/2} ∈ P and 4SAT-PROB_{> 1/2} ∈ NP-complete; but the methods used to prove these striking results stay silent about, say, 4SAT-PROB_{> 3/4}, leaving the computational complexity of kSAT-PROB_{> δ} open for most k and δ. In the present paper we give a complete characterization in the form of a trichotomy: kSAT-PROB_{> δ} lies in AC⁰, is NL-complete, or is NP-complete; and given k and δ we can decide which of the three applies. The proof of the trichotomy hinges on a new order-theoretic insight: Every set of kCNF formulas contains a formula of maximal satisfaction probability. This deceptively simple result allows us to (1) kernelize kSAT-PROB_{≥ δ}, (2) show that the variables of the kernel form a strong backdoor set when the trichotomy states membership in AC⁰ or NL, and (3) prove a locality property by which for every kCNF formula ϕ we have σ(ϕ) ≥ δ iff σ(ψ) ≥ δ for every fixed-size subset ψ of ϕ’s clauses. The locality property will allow us to prove a conjecture of Akmal and Williams: The majority-of-majority satisfaction problem for kCNFS lies in P for all k.
Cite as
Till Tantau. On the Satisfaction Probability of k-CNF Formulas. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 2:1-2:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{tantau:LIPIcs.CCC.2022.2,
author = {Tantau, Till},
title = {{On the Satisfaction Probability of k-CNF Formulas}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {2:1--2:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.2},
URN = {urn:nbn:de:0030-drops-165648},
doi = {10.4230/LIPIcs.CCC.2022.2},
annote = {Keywords: Satisfaction probability, majority it\{k\}-sat, kernelization, well orderings, locality}
}
Document
Hitting Sets for Regular Branching Programs
Abstract
We construct improved hitting set generators (HSGs) for ordered (read-once) regular branching programs in two parameter regimes. First, we construct an explicit ε-HSG for unbounded-width regular branching programs with a single accept state with seed length Õ(log n ⋅ log(1/ε)), where n is the length of the program. Second, we construct an explicit ε-HSG for width-w length-n regular branching programs with seed length Õ(log n ⋅ (√{log(1/ε)} + log w) + log(1/ε)). For context, the "baseline" in this area is the pseudorandom generator (PRG) by Nisan (Combinatorica 1992), which fools ordered (possibly non-regular) branching programs with seed length O(log(wn/ε) ⋅ log n). For regular programs, the state-of-the-art PRG, by Braverman, Rao, Raz, and Yehudayoff (FOCS 2010, SICOMP 2014), has seed length Õ(log(w/ε) ⋅ log n), which beats Nisan’s seed length when log(w/ε) = o(log n). Taken together, our two new constructions beat Nisan’s seed length in all parameter regimes except when log w and log(1/ε) are both Ω(log n) (for the construction of HSGs for regular branching programs with a single accept vertex).
Extending work by Reingold, Trevisan, and Vadhan (STOC 2006), we furthermore show that an explicit HSG for regular branching programs with a single accept vertex with seed length o(log² n) in the regime log w = Θ(log(1/ε)) = Θ(log n) would imply improved HSGs for general ordered branching programs, which would be a major breakthrough in derandomization. Pyne and Vadhan (CCC 2021) recently obtained such parameters for the special case of permutation branching programs.
Cite as
Andrej Bogdanov, William M. Hoza, Gautam Prakriya, and Edward Pyne. Hitting Sets for Regular Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bogdanov_et_al:LIPIcs.CCC.2022.3,
author = {Bogdanov, Andrej and Hoza, William M. and Prakriya, Gautam and Pyne, Edward},
title = {{Hitting Sets for Regular Branching Programs}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {3:1--3:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.3},
URN = {urn:nbn:de:0030-drops-165658},
doi = {10.4230/LIPIcs.CCC.2022.3},
annote = {Keywords: Pseudorandomness, hitting set generators, space-bounded computation}
}
Document
Linear Branching Programs and Directional Affine Extractors
Abstract
A natural model of read-once linear branching programs is a branching program where queries are 𝔽₂ linear forms, and along each path, the queries are linearly independent. We consider two restrictions of this model, which we call weakly and strongly read-once, both generalizing standard read-once branching programs and parity decision trees. Our main results are as follows.
- Average-case complexity. We define a pseudo-random class of functions which we call directional affine extractors, and show that these functions are hard on average for the strongly read-once model. We then present an explicit construction of such function with good parameters. This strengthens the result of Cohen and Shinkar (ITCS'16) who gave such average-case hardness for parity decision trees. Directional affine extractors are stronger than the more familiar class of affine extractors. Given the significance of these functions, we expect that our new class of functions might be of independent interest.
- Proof complexity. We also consider the proof system Res[⊕], which is an extension of resolution with linear queries, and define the regular variant of Res[⊕]. A refutation of a CNF in this proof system naturally defines a linear branching program solving the corresponding search problem. If a refutation is regular, we prove that the resulting program is read-once. Conversely, we show that a weakly read-once linear BP solving the search problem can be converted to a regular Res[⊕] refutation with constant blow up, where the regularity condition comes from the definition of weakly read-once BPs, thus obtaining the equivalence between these proof systems.
Cite as
Svyatoslav Gryaznov, Pavel Pudlák, and Navid Talebanfard. Linear Branching Programs and Directional Affine Extractors. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{gryaznov_et_al:LIPIcs.CCC.2022.4,
author = {Gryaznov, Svyatoslav and Pudl\'{a}k, Pavel and Talebanfard, Navid},
title = {{Linear Branching Programs and Directional Affine Extractors}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {4:1--4:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.4},
URN = {urn:nbn:de:0030-drops-165664},
doi = {10.4230/LIPIcs.CCC.2022.4},
annote = {Keywords: Boolean Functions, Average-Case Lower Bounds, AC0\lbrack2\rbrack, Affine Dispersers, Affine Extractors}
}
Document
Quantum Search-To-Decision Reductions and the State Synthesis Problem
Abstract
It is a useful fact in classical computer science that many search problems are reducible to decision problems; this has led to decision problems being regarded as the de facto computational task to study in complexity theory. In this work, we explore search-to-decision reductions for quantum search problems, wherein a quantum algorithm makes queries to a classical decision oracle to output a desired quantum state. In particular, we focus on search-to-decision reductions for QMA, and show that there exists a quantum polynomial-time algorithm that can generate a witness for a QMA problem up to inverse polynomial precision by making one query to a PP decision oracle. We complement this result by showing that QMA-search does not reduce to QMA-decision in polynomial-time, relative to a quantum oracle.
We also explore the more general state synthesis problem, in which the goal is to efficiently synthesize a target state by making queries to a classical oracle encoding the state. We prove that there exists a classical oracle with which any quantum state can be synthesized to inverse polynomial precision using only one oracle query and to inverse exponential precision using two oracle queries. This answers an open question of Aaronson [Aaronson, 2016], who presented a state synthesis algorithm that makes O(n) queries to a classical oracle to prepare an n-qubit state, and asked if the query complexity could be made sublinear.
Cite as
Sandy Irani, Anand Natarajan, Chinmay Nirkhe, Sujit Rao, and Henry Yuen. Quantum Search-To-Decision Reductions and the State Synthesis Problem. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{irani_et_al:LIPIcs.CCC.2022.5,
author = {Irani, Sandy and Natarajan, Anand and Nirkhe, Chinmay and Rao, Sujit and Yuen, Henry},
title = {{Quantum Search-To-Decision Reductions and the State Synthesis Problem}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {5:1--5:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.5},
URN = {urn:nbn:de:0030-drops-165674},
doi = {10.4230/LIPIcs.CCC.2022.5},
annote = {Keywords: Search-to-decision, state synthesis, quantum computing}
}
Document
Almost Polynomial Factor Inapproximability for Parameterized k-Clique
Abstract
The k-Clique problem is a canonical hard problem in parameterized complexity. In this paper, we study the parameterized complexity of approximating the k-Clique problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a clique of size at least k/F(k) whenever the graph G has a clique of size k. When such an algorithm runs in time T(k) ⋅ poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for the k-Clique problem.
Although, the non-existence of an F(k)-FPT-approximation algorithm for any computable sublinear function F is known under gap-ETH [Chalermsook et al., FOCS 2017], it has remained a long standing open problem to prove the same inapproximability result under the more standard and weaker assumption, W[1]≠FPT.
In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e., F(k) = O(1)) FPT-approximation algorithms under W[1]≠FPT. In this paper, we improve this inapproximability result (under the same assumption) to rule out every F(k) = k^{1/H(k)} factor FPT-approximation algorithm for any increasing computable function H (for example H(k) = log^∗ k).
Our main technical contribution is introducing list decoding of Hadamard codes over large prime fields into the proof framework of Lin.
Cite as
Karthik C. S. and Subhash Khot. Almost Polynomial Factor Inapproximability for Parameterized k-Clique. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{karthikc.s._et_al:LIPIcs.CCC.2022.6,
author = {Karthik C. S. and Khot, Subhash},
title = {{Almost Polynomial Factor Inapproximability for Parameterized k-Clique}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {6:1--6:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.6},
URN = {urn:nbn:de:0030-drops-165680},
doi = {10.4230/LIPIcs.CCC.2022.6},
annote = {Keywords: Parameterized Complexity, k-clique, Hardness of Approximation}
}
Document
𝓁_p-Spread and Restricted Isometry Properties of Sparse Random Matrices
Abstract
Random subspaces X of ℝⁿ of dimension proportional to n are, with high probability, well-spread with respect to the 𝓁₂-norm. Namely, every nonzero x ∈ X is "robustly non-sparse" in the following sense: x is ε ‖x‖₂-far in 𝓁₂-distance from all δ n-sparse vectors, for positive constants ε, δ bounded away from 0. This "𝓁₂-spread" property is the natural counterpart, for subspaces over the reals, of the minimum distance of linear codes over finite fields, and corresponds to X being a Euclidean section of the 𝓁₁ unit ball. Explicit 𝓁₂-spread subspaces of dimension Ω(n), however, are unknown, and the best known explicit constructions (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of certain sparse matrices.
Motivated by this, we study the spread properties of the kernels of sparse random matrices. We prove that with high probability such subspaces contain vectors x that are o(1)⋅‖x‖₂-close to o(n)-sparse with respect to the 𝓁₂-norm, and in particular are not 𝓁₂-spread. This is strikingly different from the case of random LDPC codes, whose distance is asymptotically almost as good as that of (dense) random linear codes.
On the other hand, for p < 2 we prove that such subspaces are 𝓁_p-spread with high probability. The spread property of sparse random matrices thus exhibits a threshold behavior at p = 2. Our proof for p < 2 moreover shows that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the 𝓁_p norm, and in fact this follows solely from the unique expansion of a random biregular graph, yielding a somewhat unexpected generalization of a similar result for the 𝓁₁ norm [Berinde et al., 2008]. Instantiating this with suitable explicit expanders, we obtain the first explicit constructions of 𝓁_p-RIP matrices for 1 ≤ p < p₀, where 1 < p₀ < 2 is an absolute constant.
Cite as
Venkatesan Guruswami, Peter Manohar, and Jonathan Mosheiff. 𝓁_p-Spread and Restricted Isometry Properties of Sparse Random Matrices. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{guruswami_et_al:LIPIcs.CCC.2022.7,
author = {Guruswami, Venkatesan and Manohar, Peter and Mosheiff, Jonathan},
title = {{𝓁\underlinep-Spread and Restricted Isometry Properties of Sparse Random Matrices}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {7:1--7:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.7},
URN = {urn:nbn:de:0030-drops-165695},
doi = {10.4230/LIPIcs.CCC.2022.7},
annote = {Keywords: Spread Subspaces, Euclidean Sections, Restricted Isometry Property, Sparse Matrices}
}
Document
Trading Time and Space in Catalytic Branching Programs
Abstract
An m-catalytic branching program (Girard, Koucký, McKenzie 2015) is a set of m distinct branching programs for f which are permitted to share internal (i.e. non-source non-sink) nodes. While originally introduced as a non-uniform analogue to catalytic space, this also gives a natural notion of amortized non-uniform space complexity for f, namely the smallest value |G|/m for an m-catalytic branching program G for f (Potechin 2017).
Potechin (2017) showed that every function f has amortized size O(n), witnessed by an m-catalytic branching program where m = 2^(2ⁿ-1). We recreate this result by defining a catalytic algorithm for evaluating polynomials using a large amount of space but O(n) time. This allows us to balance this with previously known algorithms which are efficient with respect to space at the cost of time (Cook, Mertz 2020, 2021). We show that for any ε ≥ 2n^(-1), every function f has an m-catalytic branching program of size O_ε(mn), where m = 2^(2^(ε n)). We similarly recreate an improved result due to Robere and Zuiddam (2021), and show that for d ≤ n and ε ≥ 2d^(-1), the same result holds for m = 2^binom(n, ≤ ε d) as long as f is a degree-d polynomial over 𝔽₂. We also show that for certain classes of functions, m can be reduced to 2^(poly n) while still maintaining linear or quasi-linear amortized size.
In the other direction, we bound the necessary length, and by extension the amortized size, of any permutation branching program for an arbitrary function between 3n and 4n-4.
Cite as
James Cook and Ian Mertz. Trading Time and Space in Catalytic Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{cook_et_al:LIPIcs.CCC.2022.8,
author = {Cook, James and Mertz, Ian},
title = {{Trading Time and Space in Catalytic Branching Programs}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {8:1--8:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.8},
URN = {urn:nbn:de:0030-drops-165708},
doi = {10.4230/LIPIcs.CCC.2022.8},
annote = {Keywords: complexity theory, branching programs, amortized, space complexity, catalytic computation}
}
Document
Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors
Abstract
Since the seminal works of Strassen and Valiant it has been a central theme in algebraic complexity theory to understand the relative complexity of algebraic problems, that is, to understand which algebraic problems (be it bilinear maps like matrix multiplication in Strassen’s work, or the determinant and permanent polynomials in Valiant’s) can be reduced to each other (under the appropriate notion of reduction).
In this paper we work in the setting of bilinear maps and with the usual notion of reduction that allows applying linear maps to the inputs and output of a bilinear map in order to compute another bilinear map. As our main result we determine precisely how many independent scalar multiplications can be reduced to a given bilinear map (this number is called the subrank, and extends the concept of matrix diagonalization to tensors), for essentially all (i.e. generic) bilinear maps. Namely, we prove for a generic bilinear map T : V × V → V where dim(V) = n that θ(√n) independent scalar multiplications can be reduced to T. Our result significantly improves on the previous upper bound from the work of Strassen (1991) and Bürgisser (1990) which was n^{2/3 + o(1)}. Our result is very precise and tight up to an additive constant. Our full result is much more general and applies not only to bilinear maps and 3-tensors but also to k-tensors, for which we find that the generic subrank is θ(n^{1/(k-1)}). Moreover, as an application we prove that the subrank is not additive under the direct sum.
The subrank plays a central role in several areas of complexity theory (matrix multiplication algorithms, barrier results) and combinatorics (e.g., the cap set problem and sunflower problem). As a consequence of our result we obtain several large separations between the subrank and tensor methods that have received much interest recently, notably the slice rank (Tao, 2016), analytic rank (Gowers-Wolf, 2011; Lovett, 2018; Bhrushundi-Harsha-Hatami-Kopparty-Kumar, 2020), geometric rank (Kopparty-Moshkovitz-Zuiddam, 2020), and G-stable rank (Derksen, 2020).
Our proofs of the lower bounds rely on a new technical result about an optimal decomposition of tensor space into structured subspaces, which we think may be of independent interest.
Cite as
Harm Derksen, Visu Makam, and Jeroen Zuiddam. Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 9:1-9:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{derksen_et_al:LIPIcs.CCC.2022.9,
author = {Derksen, Harm and Makam, Visu and Zuiddam, Jeroen},
title = {{Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {9:1--9:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.9},
URN = {urn:nbn:de:0030-drops-165716},
doi = {10.4230/LIPIcs.CCC.2022.9},
annote = {Keywords: tensors, bilinear maps, complexity, subrank, diagonalization, generic tensors, random tensors, reduction, slice rank}
}
Document
New Near-Linear Time Decodable Codes Closer to the GV Bound
Abstract
We construct a family of binary codes of relative distance 1/2-ε and rate ε² ⋅ 2^(-log^α (1/ε)) for α ≈ 1/2 that are decodable, probabilistically, in near-linear time. This improves upon the rate of the state-of-the-art near-linear time decoding near the GV bound due to Jeronimo, Srivastava, and Tulsiani, who gave a randomized decoding of Ta-Shma codes with α ≈ 5/6 [Ta-Shma, 2017; Jeronimo et al., 2021]. Each code in our family can be constructed in probabilistic polynomial time, or deterministic polynomial time given sufficiently good explicit 3-uniform hypergraphs.
Our construction is based on a new graph-based bias amplification method. While previous works start with some base code of relative distance 1/2-ε₀ for ε₀ ≫ ε and amplify the distance to 1/2-ε by walking on an expander, or on a carefully tailored product of expanders, we walk over very sparse, highly mixing, hypergraphs. Study of such hypergraphs further offers an avenue toward achieving rate Ω̃(ε²). For our unique- and list-decoding algorithms, we employ the framework developed in [Jeronimo et al., 2021].
Cite as
Guy Blanc and Dean Doron. New Near-Linear Time Decodable Codes Closer to the GV Bound. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 10:1-10:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{blanc_et_al:LIPIcs.CCC.2022.10,
author = {Blanc, Guy and Doron, Dean},
title = {{New Near-Linear Time Decodable Codes Closer to the GV Bound}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {10:1--10:40},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.10},
URN = {urn:nbn:de:0030-drops-165726},
doi = {10.4230/LIPIcs.CCC.2022.10},
annote = {Keywords: Unique decoding, list decoding, the Gilbert-Varshamov bound, small-bias sample spaces, hypergraphs, expander walks}
}
Document
Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance
Abstract
An active topic in the study of random constraint satisfaction problems (CSPs) is the geometry of the space of satisfying or almost satisfying assignments as the function of the density, for which a precise landscape of predictions has been made via statistical physics-based heuristics. In parallel, there has been a recent flurry of work on refuting random constraint satisfaction problems, via nailing refutation thresholds for spectral and semidefinite programming-based algorithms, and also on counting solutions to CSPs. Inspired by this, the starting point for our work is the following question: What does the solution space for a random CSP look like to an efficient algorithm?
In pursuit of this inquiry, we focus on the following problems about random Boolean CSPs at the densities where they are unsatisfiable but no refutation algorithm is known.
1) Counts. For every Boolean CSP we give algorithms that with high probability certify a subexponential upper bound on the number of solutions. We also give algorithms to certify a bound on the number of large cuts in a Gaussian-weighted graph, and the number of large independent sets in a random d-regular graph.
2) Clusters. For Boolean 3CSPs we give algorithms that with high probability certify an upper bound on the number of clusters of solutions.
3) Balance. We also give algorithms that with high probability certify that there are no "unbalanced" solutions, i.e., solutions where the fraction of +1s deviates significantly from 50%. Finally, we also provide hardness evidence suggesting that our algorithms for counting are optimal.
Cite as
Jun-Ting Hsieh, Sidhanth Mohanty, and Jeff Xu. Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{hsieh_et_al:LIPIcs.CCC.2022.11,
author = {Hsieh, Jun-Ting and Mohanty, Sidhanth and Xu, Jeff},
title = {{Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {11:1--11:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.11},
URN = {urn:nbn:de:0030-drops-165735},
doi = {10.4230/LIPIcs.CCC.2022.11},
annote = {Keywords: constraint satisfaction problems, certified counting, random graphs}
}
Document
On Efficient Noncommutative Polynomial Factorization via Higman Linearization
Abstract
In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring 𝔽∠{x_1,x_2,…,x_n} of polynomials in noncommuting variables x_1,x_2,…,x_n over the field 𝔽. We obtain the following result:
- We give a randomized algorithm that takes as input a noncommutative arithmetic formula of size s computing a noncommutative polynomial f ∈ 𝔽∠{x_1,x_2,…,x_n}, where 𝔽 = 𝔽_q is a finite field, and in time polynomial in s, n and log₂q computes a factorization of f as a product f = f_1f_2 ⋯ f_r, where each f_i is an irreducible polynomial that is output as a noncommutative algebraic branching program.
- The algorithm works by first transforming f into a linear matrix L using Higman’s linearization of polynomials. We then factorize the linear matrix L and recover the factorization of f. We use basic elements from Cohn’s theory of free ideals rings combined with Ronyai’s randomized polynomial-time algorithm for computing invariant subspaces of a collection of matrices over finite fields.
Cite as
Vikraman Arvind and Pushkar S. Joglekar. On Efficient Noncommutative Polynomial Factorization via Higman Linearization. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{arvind_et_al:LIPIcs.CCC.2022.12,
author = {Arvind, Vikraman and Joglekar, Pushkar S.},
title = {{On Efficient Noncommutative Polynomial Factorization via Higman Linearization}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {12:1--12:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.12},
URN = {urn:nbn:de:0030-drops-165747},
doi = {10.4230/LIPIcs.CCC.2022.12},
annote = {Keywords: Noncommutative Polynomials, Arithmetic Circuits, Factorization, Identity testing}
}
Document
A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates
Abstract
We prove that a modification of Andreev’s function is not computable by (3 + α - ε) log(n) depth De Morgan formula with (2α - ε)log{n} layers of AND gates at the top for any 0 < α < 1/5 and any constant ε > 0. In order to do this, we prove a weak variant of Karchmer-Raz-Wigderson conjecture. To be more precise, we prove the existence of two functions f : {0,1}ⁿ → {0,1} and g : {0,1}ⁿ → {0,1}ⁿ such that f(g(x) ⊕ y) is not computable by depth (1 + α - ε) n formulas with (2 α - ε) n layers of AND gates at the top. We do this by a top-down approach, which was only used before for depth-3 model.
Our technical contribution includes combinatorial insights into structure of composition with random boolean function, which led us to introducing a notion of well-mixed sets. A set of functions is well-mixed if, when composed with a random function, it does not have subsets that agree on large fractions of inputs. We use probabilistic method to prove the existence of well-mixed sets.
Cite as
Ivan Mihajlin and Anastasia Sofronova. A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{mihajlin_et_al:LIPIcs.CCC.2022.13,
author = {Mihajlin, Ivan and Sofronova, Anastasia},
title = {{A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {13:1--13:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.13},
URN = {urn:nbn:de:0030-drops-165755},
doi = {10.4230/LIPIcs.CCC.2022.13},
annote = {Keywords: formula complexity, communication complexity, Karchmer-Raz-Wigderson conjecture, De Morgan formulas}
}
Document
The Plane Test Is a Local Tester for Multiplicity Codes
Abstract
Multiplicity codes are a generalization of RS and RM codes where for each evaluation point we output the evaluation of a low-degree polynomial and all of its directional derivatives up to order s. Multi-variate multiplicity codes are locally decodable with the natural local decoding algorithm that reads values on a random line and corrects to the closest uni-variate multiplicity code. However, it was not known whether multiplicity codes are locally testable, and this question has been posed since the introduction of these codes with no progress up to date. In fact, it has been also open whether multiplicity codes can be characterized by local constraints, i.e., if there exists a probabilistic algorithm that queries few symbols of a word c, accepts every c in the code with probability 1, and rejects every c not in the code with nonzero probability.
We begin by giving a simple example showing the line test does not give local characterization when d > q. Surprisingly, we then show the plane test is a local characterization when s < q and d < qs-1 for prime q. In addition, we show the s-dimensional test is a local tester for multiplicity codes, when s < q. Combining the two results, we show our main result that the plane test is a local tester for multiplicity codes of degree d < qs-1, with constant rejection probability for constant q, s.
Our technique is new. We represent the given input as a possibly very high-degree polynomial, and we show that for some choice of plane, the restriction of the polynomial to the plane is a high-degree bi-variate polynomial. The argument has to work modulo the appropriate kernels, and for that we use Grobner theory, the Combinatorial Nullstellensatz theorem and its generalization to multiplicities. Even given that, the argument is delicate and requires choosing a non-standard monomial order for the argument to work.
Cite as
Dan Karliner, Roie Salama, and Amnon Ta-Shma. The Plane Test Is a Local Tester for Multiplicity Codes. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 14:1-14:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{karliner_et_al:LIPIcs.CCC.2022.14,
author = {Karliner, Dan and Salama, Roie and Ta-Shma, Amnon},
title = {{The Plane Test Is a Local Tester for Multiplicity Codes}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {14:1--14:33},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.14},
URN = {urn:nbn:de:0030-drops-165761},
doi = {10.4230/LIPIcs.CCC.2022.14},
annote = {Keywords: local testing, multiplicity codes, Reed Muller codes}
}
Document
Pseudorandom Generators, Resolution and Heavy Width
Abstract
Following the paper of Alekhnovich, Ben-Sasson, Razborov, Wigderson [Michael Alekhnovich et al., 2004] we call a pseudorandom generator ℱ:{0, 1}ⁿ → {0, 1}^m hard for a propositional proof system P if P cannot efficiently prove the (properly encoded) statement b ∉ Im(ℱ) for any string b ∈ {0, 1}^m.
In [Michael Alekhnovich et al., 2004] the authors suggested the "functional encoding" of the considered statement for Nisan-Wigderson generator that allows the introduction of "local" extension variables. These extension variables may potentially significantly increase the power of the proof system. In [Michael Alekhnovich et al., 2004] authors gave a lower bound of exp[Ω(n²/{m⋅2^{2^Δ}})] on the length of Resolution proofs where Δ is the degree of the dependency graph of the generator. This lower bound meets the barrier for the restriction technique.
In this paper, we introduce a "heavy width" measure for Resolution that allows us to show a lower bound of exp[n²/{m 2^𝒪(εΔ)}] on the length of Resolution proofs of the considered statement for the Nisan-Wigderson generator. This gives an exponential lower bound up to Δ := log^{2 - δ} n (the bigger degree the more extension variables we can use). In [Michael Alekhnovich et al., 2004] authors left an open problem to get rid of scaling factor 2^{2^Δ}, it is a solution to this open problem.
Cite as
Dmitry Sokolov. Pseudorandom Generators, Resolution and Heavy Width. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{sokolov:LIPIcs.CCC.2022.15,
author = {Sokolov, Dmitry},
title = {{Pseudorandom Generators, Resolution and Heavy Width}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {15:1--15:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.15},
URN = {urn:nbn:de:0030-drops-165770},
doi = {10.4230/LIPIcs.CCC.2022.15},
annote = {Keywords: proof complexity, pseudorandom generators, resolution, lower bounds}
}
Document
Probabilistic Kolmogorov Complexity with Applications to Average-Case Complexity
Abstract
Understanding the relationship between the worst-case and average-case complexities of NP and of other subclasses of PH is a long-standing problem in complexity theory. Over the last few years, much progress has been achieved in this front through the investigation of meta-complexity: the complexity of problems that refer to the complexity of the input string x (e.g., given a string x, estimate its time-bounded Kolmogorov complexity). In particular, [Shuichi Hirahara, 2021] employed techniques from meta-complexity to show that if DistNP ⊆ AvgP then UP ⊆ DTIME[2^{O(n/log n)}]. While this and related results [Shuichi Hirahara and Mikito Nanashima, 2021; Lijie Chen et al., 2022] offer exciting progress after a long gap, they do not survive in the setting of randomized computations: roughly speaking, "randomness" is the opposite of "structure", and upper bounding the amount of structure (time-bounded Kolmogorov complexity) of different objects is crucial in recent applications of meta-complexity. This limitation is significant, since randomized computations are ubiquitous in algorithm design and give rise to a more robust theory of average-case complexity [Russell Impagliazzo and Leonid A. Levin, 1990].
In this work, we develop a probabilistic theory of meta-complexity, by incorporating randomness into the notion of complexity of a string x. This is achieved through a new probabilistic variant of time-bounded Kolmogorov complexity that we call pK^t complexity. Informally, pK^t(x) measures the complexity of x when shared randomness is available to all parties involved in a computation. By porting key results from meta-complexity to the probabilistic domain of pK^t complexity and its variants, we are able to establish new connections between worst-case and average-case complexity in the important setting of probabilistic computations:
- If DistNP ⊆ AvgBPP, then UP ⊆ RTIME[2^O(n/log n)].
- If DistΣ^P_2 ⊆ AvgBPP, then AM ⊆ BPTIME[2^O(n/log n)].
- In the fine-grained setting [Lijie Chen et al., 2022], we get UTIME[2^O(√{nlog n})] ⊆ RTIME[2^O(√{nlog n})] and AMTIME[2^O(√{nlog n})] ⊆ BPTIME[2^O(√{nlog n})] from stronger average-case assumptions.
- If DistPH ⊆ AvgBPP, then PH ⊆ BPTIME[2^O(n/log n)]. Specifically, for any 𝓁 ≥ 0, if DistΣ_{𝓁+2}^P ⊆ AvgBPP then Σ_𝓁^{P} ⊆ BPTIME[2^O(n/log n)].
- Strengthening a result from [Shuichi Hirahara and Mikito Nanashima, 2021], we show that if DistNP ⊆ AvgBPP then polynomial size Boolean circuits can be agnostically PAC learned under any unknown 𝖯/poly-samplable distribution in polynomial time. In some cases, our framework allows us to significantly simplify existing proofs, or to extend results to the more challenging probabilistic setting with little to no extra effort.
Cite as
Halley Goldberg, Valentine Kabanets, Zhenjian Lu, and Igor C. Oliveira. Probabilistic Kolmogorov Complexity with Applications to Average-Case Complexity. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 16:1-16:60, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{goldberg_et_al:LIPIcs.CCC.2022.16,
author = {Goldberg, Halley and Kabanets, Valentine and Lu, Zhenjian and Oliveira, Igor C.},
title = {{Probabilistic Kolmogorov Complexity with Applications to Average-Case Complexity}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {16:1--16:60},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.16},
URN = {urn:nbn:de:0030-drops-165785},
doi = {10.4230/LIPIcs.CCC.2022.16},
annote = {Keywords: average-case complexity, Kolmogorov complexity, meta-complexity, worst-case to average-case reductions, learning}
}
Document
Nisan-Wigderson Generators in Proof Complexity: New Lower Bounds
Abstract
A map g:{0,1}ⁿ → {0,1}^m (m > n) is a hard proof complexity generator for a proof system P iff for every string b ∈ {0,1}^m ⧵ Rng(g), formula τ_b(g) naturally expressing b ∉ Rng(g) requires superpolynomial size P-proofs. One of the well-studied maps in the theory of proof complexity generators is Nisan-Wigderson generator. Razborov [A. A. {Razborov}, 2015] conjectured that if A is a suitable matrix and f is a NP∩CoNP function hard-on-average for 𝖯/poly, then NW_{f, A} is a hard proof complexity generator for Extended Frege. In this paper, we prove a form of Razborov’s conjecture for AC⁰-Frege. We show that for any symmetric NP∩CoNP function f that is exponentially hard for depth two AC⁰ circuits, NW_{f,A} is a hard proof complexity generator for AC⁰-Frege in a natural setting. As direct applications of this theorem, we show that:
1) For any f with the specified properties, τ_b(NW_{f,A}) (for a natural formalization) based on a random b and a random matrix A with probability 1-o(1) is a tautology and requires superpolynomial (or even exponential) AC⁰-Frege proofs.
2) Certain formalizations of the principle f_n ∉ (NP∩CoNP)/poly requires superpolynomial AC⁰-Frege proofs. These applications relate to two questions that were asked by Krajíček [J. {Krajíček}, 2019].
Cite as
Erfan Khaniki. Nisan-Wigderson Generators in Proof Complexity: New Lower Bounds. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{khaniki:LIPIcs.CCC.2022.17,
author = {Khaniki, Erfan},
title = {{Nisan-Wigderson Generators in Proof Complexity: New Lower Bounds}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {17:1--17:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.17},
URN = {urn:nbn:de:0030-drops-165799},
doi = {10.4230/LIPIcs.CCC.2022.17},
annote = {Keywords: Proof complexity, Bounded arithmetic, Bounded depth Frege, Nisan-Wigderson generators, Meta-complexity, Lower bounds}
}
Document
High-Dimensional Expanders from Chevalley Groups
Abstract
Let Φ be an irreducible root system (other than G₂) of rank at least 2, let 𝔽 be a finite field with p = char 𝔽 > 3, and let G(Φ,𝔽) be the corresponding Chevalley group. We describe a strongly explicit high-dimensional expander (HDX) family of dimension rank(Φ), where G(Φ,𝔽) acts simply transitively on the top-dimensional faces; these are λ-spectral HDXs with λ → 0 as p → ∞. This generalizes a construction of Kaufman and Oppenheim (STOC 2018), which corresponds to the case Φ = A_d. Our work gives three new families of spectral HDXs of any dimension ≥ 2, and four exceptional constructions of dimension 4, 6, 7, and 8.
Cite as
Ryan O'Donnell and Kevin Pratt. High-Dimensional Expanders from Chevalley Groups. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 18:1-18:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{odonnell_et_al:LIPIcs.CCC.2022.18,
author = {O'Donnell, Ryan and Pratt, Kevin},
title = {{High-Dimensional Expanders from Chevalley Groups}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {18:1--18:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.18},
URN = {urn:nbn:de:0030-drops-165802},
doi = {10.4230/LIPIcs.CCC.2022.18},
annote = {Keywords: High-dimensional expanders, simplicial complexes, group theory}
}
Document
The Composition Complexity of Majority
Abstract
We study the complexity of computing majority as a composition of local functions: Maj_n = h(g_1,…,g_m), where each g_j: {0,1}ⁿ → {0,1} is an arbitrary function that queries only k ≪ n variables and h: {0,1}^m → {0,1} is an arbitrary combining function. We prove an optimal lower bound of m ≥ Ω(n/k log k) on the number of functions needed, which is a factor Ω(log k) larger than the ideal m = n/k. We call this factor the composition overhead; previously, no superconstant lower bounds on it were known for majority.
Our lower bound recovers, as a corollary and via an entirely different proof, the best known lower bound for bounded-width branching programs for majority (Alon and Maass '86, Babai et al. '90). It is also the first step in a plan that we propose for breaking a longstanding barrier in lower bounds for small-depth boolean circuits.
Novel aspects of our proof include sharp bounds on the information lost as computation flows through the inner functions g_j, and the bootstrapping of lower bounds for a multi-output function (Hamming weight) into lower bounds for a single-output one (majority).
Cite as
Victor Lecomte, Prasanna Ramakrishnan, and Li-Yang Tan. The Composition Complexity of Majority. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 19:1-19:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{lecomte_et_al:LIPIcs.CCC.2022.19,
author = {Lecomte, Victor and Ramakrishnan, Prasanna and Tan, Li-Yang},
title = {{The Composition Complexity of Majority}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {19:1--19:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.19},
URN = {urn:nbn:de:0030-drops-165818},
doi = {10.4230/LIPIcs.CCC.2022.19},
annote = {Keywords: computational complexity, circuit lower bounds}
}
Document
The Acrobatics of BQP
Abstract
One can fix the randomness used by a randomized algorithm, but there is no analogous notion of fixing the quantumness used by a quantum algorithm. Underscoring this fundamental difference, we show that, in the black-box setting, the behavior of quantum polynomial-time (BQP) can be remarkably decoupled from that of classical complexity classes like NP. Specifically:
- There exists an oracle relative to which NP^{BQP} ⊄ BQP^{PH}, resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which 𝖯 = NP but BQP ≠ QCMA.
- Conversely, there exists an oracle relative to which BQP^{NP} ⊄ PH^{BQP}.
- Relative to a random oracle, PP is not contained in the "QMA hierarchy" QMA^{QMA^{QMA^{⋯}}}.
- Relative to a random oracle, Σ_{k+1}^𝖯 ⊄ BQP^{Σ_k^𝖯} for every k.
- There exists an oracle relative to which BQP = P^#P and yet PH is infinite. (By contrast, relative to all oracles, if NP ⊆ BPP, then PH collapses.)
- There exists an oracle relative to which 𝖯 = NP ≠ BQP = 𝖯^#P.
To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which BQP ⊄ PH, and associated results about the Forrelation problem. We also introduce new tools that might be of independent interest. These include a "quantum-aware" version of the random restriction method, a concentration theorem for the block sensitivity of AC⁰ circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.
Cite as
Scott Aaronson, DeVon Ingram, and William Kretschmer. The Acrobatics of BQP. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{aaronson_et_al:LIPIcs.CCC.2022.20,
author = {Aaronson, Scott and Ingram, DeVon and Kretschmer, William},
title = {{The Acrobatics of BQP}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {20:1--20:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.20},
URN = {urn:nbn:de:0030-drops-165820},
doi = {10.4230/LIPIcs.CCC.2022.20},
annote = {Keywords: BQP, Forrelation, oracle separations, Polynomial Hierarchy, query complexity}
}
Document
Random Restrictions and PRGs for PTFs in Gaussian Space
Abstract
A polynomial threshold function (PTF) f: ℝⁿ → ℝ is a function of the form f(x) = sign(p(x)) where p is a polynomial of degree at most d. PTFs are a classical and well-studied complexity class with applications across complexity theory, learning theory, approximation theory, quantum complexity and more. We address the question of designing pseudorandom generators (PRGs) for polynomial threshold functions (PTFs) in the gaussian space: design a PRG that takes a seed of few bits of randomness and outputs a n-dimensional vector whose distribution is indistinguishable from a standard multivariate gaussian by a degree d PTF.
Our main result is a PRG that takes a seed of d^O(1) log(n/ε) log(1/ε)/ε² random bits with output that cannot be distinguished from an n-dimensional gaussian distribution with advantage better than ε by degree d PTFs. The best previous generator due to O'Donnell, Servedio, and Tan (STOC'20) had a quasi-polynomial dependence (i.e., seed length of d^O(log d)) in the degree d. Along the way we prove a few nearly-tight structural properties of restrictions of PTFs that may be of independent interest.
Similar results were obtained in [Ryan O'Donnell et al., 2021] (independently and concurrently).
Cite as
Zander Kelley and Raghu Meka. Random Restrictions and PRGs for PTFs in Gaussian Space. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kelley_et_al:LIPIcs.CCC.2022.21,
author = {Kelley, Zander and Meka, Raghu},
title = {{Random Restrictions and PRGs for PTFs in Gaussian Space}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {21:1--21:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.21},
URN = {urn:nbn:de:0030-drops-165836},
doi = {10.4230/LIPIcs.CCC.2022.21},
annote = {Keywords: polynomial threshold function, pseudorandom generator, multivariate gaussian}
}
Document
Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation
Abstract
For any real numbers B ≥ 1 and δ ∈ (0,1) and function f: [0,B] → ℝ, let d_{B; δ}(f) ∈ ℤ_{> 0} denote the minimum degree of a polynomial p(x) satisfying sup_{x ∈ [0,B]} |p(x) - f(x)| < δ. In this paper, we provide precise asymptotics for d_{B; δ}(e^{-x}) and d_{B; δ}(e^x) in terms of both B and δ, improving both the previously known upper bounds and lower bounds. In particular, we show d_{B; δ}(e^{-x}) = Θ(max{√{B log(δ^{-1})}, log(δ^{-1})/{log(B^{-1} log(δ^{-1}))}}), and d_{B; δ}(e^{x}) = Θ(max{B, log(δ^{-1})/{log(B^{-1} log(δ^{-1}))}}), and we explicitly determine the leading coefficients in most parameter regimes.
Polynomial approximations for e^{-x} and e^x have applications to the design of algorithms for many problems, including in scientific computing, graph algorithms, machine learning, and statistics. Our degree bounds show both the power and limitations of these algorithms.
We focus in particular on the Batch Gaussian Kernel Density Estimation problem for n sample points in Θ(log n) dimensions with error δ = n^{-Θ(1)}. We show that the running time one can achieve depends on the square of the diameter of the point set, B, with a transition at B = Θ(log n) mirroring the corresponding transition in d_{B; δ}(e^{-x}):
- When B = o(log n), we give the first algorithm running in time n^{1 + o(1)}.
- When B = κ log n for a small constant κ > 0, we give an algorithm running in time n^{1 + O(log log κ^{-1} /log κ^{-1})}. The log log κ^{-1} /log κ^{-1} term in the exponent comes from analyzing the behavior of the leading constant in our computation of d_{B; δ}(e^{-x}).
- When B = ω(log n), we show that time n^{2 - o(1)} is necessary assuming SETH.
Cite as
Amol Aggarwal and Josh Alman. Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{aggarwal_et_al:LIPIcs.CCC.2022.22,
author = {Aggarwal, Amol and Alman, Josh},
title = {{Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {22:1--22:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.22},
URN = {urn:nbn:de:0030-drops-165846},
doi = {10.4230/LIPIcs.CCC.2022.22},
annote = {Keywords: polynomial approximation, kernel density estimation, Chebyshev polynomials}
}
Document
Extremely Efficient Constructions of Hash Functions, with Applications to Hardness Magnification and PRFs
Abstract
In a recent work, Fan, Li, and Yang (STOC 2022) constructed a family of almost-universal hash functions such that each function in the family is computable by (2n + o(n))-gate circuits of fan-in 2 over the B₂ basis. Applying this family, they established the existence of pseudorandom functions computable by circuits of the same complexity, under the standard assumption that OWFs exist. However, a major disadvantage of the hash family construction by Fan, Li, and Yang (STOC 2022) is that it requires a seed length of poly(n), which limits its potential applications.
We address this issue by giving an improved construction of almost-universal hash functions with seed length polylog(n), such that each function in the family is computable with POLYLOGTIME-uniform (2n + o(n))-gate circuits. Our new construction has the following applications in both complexity theory and cryptography.
- (Hardness magnification). Let α : ℕ → ℕ be any function such that α(n) ≤ log n / log log n. We show that if there is an n^{α(n)}-sparse NP language that does not have probabilistic circuits of 2n + O(n/log log n) gates, then we have (1) NTIME[2ⁿ] ⊈ SIZE[2^{n^{1/5}}] and (2) NP ⊈ SIZE[n^k] for every constant k. Complementing this magnification phenomenon, we present an O(n)-sparse language in P which requires probabilistic circuits of size at least 2n - 2. This is the first result in hardness magnification showing that even a sub-linear additive improvement on known circuit size lower bounds would imply NEXP ⊄ P_{/poly}.
Following Chen, Jin, and Williams (STOC 2020), we also establish a sharp threshold for explicit obstructions: we give an explict obstruction against (2n-2)-size circuits, and prove that a sub-linear additive improvement on the circuit size would imply (1) DTIME[2ⁿ] ⊈ SIZE[2^{n^{1/5}}] and (2) P ⊈ SIZE[n^k] for every constant k.
- (Extremely efficient construction of pseudorandom functions). Assuming that one of integer factoring, decisional Diffie-Hellman, or ring learning-with-errors is sub-exponentially hard, we show the existence of pseudorandom functions computable by POLYLOGTIME-uniform AC⁰[2] circuits with 2n + o(n) wires, with key length polylog(n). We also show that PRFs computable by POLYLOGTIME-uniform B₂ circuits of 2n + o(n) gates follows from the existence of sub-exponentially secure one-way functions.
Cite as
Lijie Chen, Jiatu Li, and Tianqi Yang. Extremely Efficient Constructions of Hash Functions, with Applications to Hardness Magnification and PRFs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 23:1-23:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chen_et_al:LIPIcs.CCC.2022.23,
author = {Chen, Lijie and Li, Jiatu and Yang, Tianqi},
title = {{Extremely Efficient Constructions of Hash Functions, with Applications to Hardness Magnification and PRFs}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {23:1--23:37},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.23},
URN = {urn:nbn:de:0030-drops-165852},
doi = {10.4230/LIPIcs.CCC.2022.23},
annote = {Keywords: Almost universal hash functions, hardness magnification, pseudorandom functions}
}
Document
Hardness of Approximation for Stochastic Problems via Interactive Oracle Proofs
Abstract
Hardness of approximation aims to establish lower bounds on the approximability of optimization problems in NP and beyond. We continue the study of hardness of approximation for problems beyond NP, specifically for stochastic constraint satisfaction problems (SCSPs). An SCSP with 𝗄 alternations is a list of constraints over variables grouped into 2𝗄 blocks, where each constraint has constant arity. An assignment to the SCSP is defined by two players who alternate in setting values to a designated block of variables, with one player choosing their assignments uniformly at random and the other player trying to maximize the number of satisfied constraints.
In this paper, we establish hardness of approximation for SCSPs based on interactive proofs. For 𝗄 ≤ O(log n), we prove that it is AM[𝗄]-hard to approximate, to within a constant, the value of SCSPs with 𝗄 alternations and constant arity. Before, this was known only for 𝗄 = O(1).
Furthermore, we introduce a natural class of 𝗄-round interactive proofs, denoted IR[𝗄] (for interactive reducibility), and show that several protocols (e.g., the sumcheck protocol) are in IR[𝗄]. Using this notion, we extend our inapproximability to all values of 𝗄: we show that for every 𝗄, approximating an SCSP instance with O(𝗄) alternations and constant arity is IR[𝗄]-hard.
While hardness of approximation for CSPs is achieved by constructing suitable PCPs, our results for SCSPs are achieved by constructing suitable IOPs (interactive oracle proofs). We show that every language in AM[𝗄 ≤ O(log n)] or in IR[𝗄] has an O(𝗄)-round IOP whose verifier has constant query complexity (regardless of the number of rounds 𝗄). In particular, we derive a "sumcheck protocol" whose verifier reads O(1) bits from the entire interaction transcript.
Cite as
Gal Arnon, Alessandro Chiesa, and Eylon Yogev. Hardness of Approximation for Stochastic Problems via Interactive Oracle Proofs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{arnon_et_al:LIPIcs.CCC.2022.24,
author = {Arnon, Gal and Chiesa, Alessandro and Yogev, Eylon},
title = {{Hardness of Approximation for Stochastic Problems via Interactive Oracle Proofs}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {24:1--24:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.24},
URN = {urn:nbn:de:0030-drops-165867},
doi = {10.4230/LIPIcs.CCC.2022.24},
annote = {Keywords: hardness of approximation, interactive oracle proofs, stochastic satisfaction problems}
}
Document
Finding Errorless Pessiland in Error-Prone Heuristica
Abstract
Average-case complexity has two standard formulations, i.e., errorless complexity and error-prone complexity. In average-case complexity, a critical topic of research is to show the equivalence between these formulations, especially on the average-case complexity of NP.
In this study, we present a relativization barrier for such an equivalence. Specifically, we construct an oracle relative to which NP is easy on average in the error-prone setting (i.e., DistNP ⊆ HeurP) but hard on average in the errorless setting even by 2^o(n/log n)-size circuits (i.e., DistNP ⊈ AvgSIZE[2^o(n/log n)]), which provides an answer to the open question posed by Impagliazzo (CCC 2011). Additionally, we show the following in the same relativized world:
- Lower bound of meta-complexity: GapMINKT^𝒪 ∉ prSIZE^𝒪[2^o(n/log n)] and GapMCSP^𝒪 ∉ prSIZE^𝒪[2^(n^ε)] for some ε > 0.
- Worst-case hardness of learning on uniform distributions: P/poly is not weakly PAC learnable with membership queries on the uniform distribution by nonuniform 2ⁿ/n^ω(1)-time algorithms.
- Average-case hardness of distribution-free learning: P/poly is not weakly PAC learnable on average by nonuniform 2^o(n/log n)-time algorithms.
- Weak cryptographic primitives: There exist a hitting set generator, an auxiliary-input one-way function, an auxiliary-input pseudorandom generator, and an auxiliary-input pseudorandom function against SIZE^𝒪[2^o(n/log n)].
This provides considerable insights into Pessiland (i.e., the world in which no one-way function exists, and NP is hard on average), such as the relativized separation of the error-prone average-case hardness of NP and auxiliary-input cryptography. At the core of our oracle construction is a new notion of random restriction with masks.
Cite as
Shuichi Hirahara and Mikito Nanashima. Finding Errorless Pessiland in Error-Prone Heuristica. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 25:1-25:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{hirahara_et_al:LIPIcs.CCC.2022.25,
author = {Hirahara, Shuichi and Nanashima, Mikito},
title = {{Finding Errorless Pessiland in Error-Prone Heuristica}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {25:1--25:28},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.25},
URN = {urn:nbn:de:0030-drops-165875},
doi = {10.4230/LIPIcs.CCC.2022.25},
annote = {Keywords: average-case complexity, oracle separation, relativization barrier, meta-complexity, learning, auxiliary-input cryptography}
}
Document
Symmetry of Information from Meta-Complexity
Abstract
Symmetry of information for time-bounded Kolmogorov complexity is a hypothetical inequality that relates time-bounded Kolmogorov complexity and its conditional analogue. In 1992, Longpré and Watanabe showed that symmetry of information holds if NP is easy in the worst case, which has been the state of the art over the last three decades. In this paper, we significantly improve this result by showing that symmetry of information holds under the weaker assumption that NP is easy on average. In fact, our proof techniques are applicable to any resource-bounded Kolmogorov complexity and enable proving symmetry of information from an efficient algorithm that computes resource-bounded Kolmogorov complexity.
We demonstrate the significance of our proof techniques by presenting two applications. First, using that symmetry of information does not hold for Levin’s Kt-complexity, we prove that randomized Kt-complexity cannot be computed in time 2^o(n) on inputs of length n, which improves the previous quasi-polynomial lower bound of Oliveira (ICALP 2019). Our proof implements Kolmogorov’s insightful approach to the P versus NP problem in the case of randomized Kt-complexity. Second, we consider the question of excluding Heuristica, i.e., a world in which NP is easy on average but NP ≠ P, from Impagliazzo’s five worlds: Using symmetry of information, we prove that Heuristica is excluded if the problem of approximating time-bounded conditional Kolmogorov complexity K^t(x∣y) up to some additive error is NP-hard for t ≫ |y|. We complement this result by proving NP-hardness of approximating sublinear-time-bounded conditional Kolmogorov complexity up to a multiplicative factor of |x|^{1/(log log |x|)^O(1)} for t ≪ |y|. Our NP-hardness proof presents a new connection between sublinear-time-bounded conditional Kolmogorov complexity and a secret sharing scheme.
Cite as
Shuichi Hirahara. Symmetry of Information from Meta-Complexity. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 26:1-26:41, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{hirahara:LIPIcs.CCC.2022.26,
author = {Hirahara, Shuichi},
title = {{Symmetry of Information from Meta-Complexity}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {26:1--26:41},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.26},
URN = {urn:nbn:de:0030-drops-165880},
doi = {10.4230/LIPIcs.CCC.2022.26},
annote = {Keywords: resource-bounded Kolmogorov complexity, average-case complexity, pseudorandomness, hardness of approximation, unconditional lower bound}
}
Document
Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs
Abstract
We study the pseudorandomness of random walks on expander graphs against tests computed by symmetric functions and permutation branching programs. These questions are motivated by applications of expander walks in the coding theory and derandomization literatures. A line of prior work has shown that random walks on expanders with second largest eigenvalue λ fool symmetric functions up to a O(λ) error in total variation distance, but only for the case where the vertices are labeled with symbols from a binary alphabet, and with a suboptimal dependence on the bias of the labeling. We generalize these results to labelings with an arbitrary alphabet, and for the case of binary labelings we achieve an optimal dependence on the labeling bias. We extend our analysis to unify it with and strengthen the expander-walk Chernoff bound. We then show that expander walks fool permutation branching programs up to a O(λ) error in 𝓁₂-distance, and we prove that much stronger bounds hold for programs with a certain structure. We also prove lower bounds to show that our results are tight. To prove our results for symmetric functions, we analyze the Fourier coefficients of the relevant distributions using linear-algebraic techniques. Our analysis for permutation branching programs is likewise linear-algebraic in nature, but also makes use of the recently introduced singular-value approximation notion for matrices (Ahmadinejad et al. 2021).
Cite as
Louis Golowich and Salil Vadhan. Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{golowich_et_al:LIPIcs.CCC.2022.27,
author = {Golowich, Louis and Vadhan, Salil},
title = {{Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {27:1--27:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.27},
URN = {urn:nbn:de:0030-drops-165893},
doi = {10.4230/LIPIcs.CCC.2022.27},
annote = {Keywords: Expander graph, Random walk, Pseudorandomness}
}
Document
Influence in Completely Bounded Block-Multilinear Forms and Classical Simulation of Quantum Algorithms
Abstract
The Aaronson-Ambainis conjecture (Theory of Computing '14) says that every low-degree bounded polynomial on the Boolean hypercube has an influential variable. This conjecture, if true, would imply that the acceptance probability of every d-query quantum algorithm can be well-approximated almost everywhere (i.e., on almost all inputs) by a poly(d)-query classical algorithm. We prove a special case of the conjecture: in every completely bounded degree-d block-multilinear form with constant variance, there always exists a variable with influence at least 1/poly(d). In a certain sense, such polynomials characterize the acceptance probability of quantum query algorithms, as shown by Arunachalam, Briët and Palazuelos (SICOMP '19). As a corollary we obtain efficient classical almost-everywhere simulation for a particular class of quantum algorithms that includes for instance k-fold Forrelation. Our main technical result relies on connections to free probability theory.
Cite as
Nikhil Bansal, Makrand Sinha, and Ronald de Wolf. Influence in Completely Bounded Block-Multilinear Forms and Classical Simulation of Quantum Algorithms. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bansal_et_al:LIPIcs.CCC.2022.28,
author = {Bansal, Nikhil and Sinha, Makrand and de Wolf, Ronald},
title = {{Influence in Completely Bounded Block-Multilinear Forms and Classical Simulation of Quantum Algorithms}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {28:1--28:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.28},
URN = {urn:nbn:de:0030-drops-165908},
doi = {10.4230/LIPIcs.CCC.2022.28},
annote = {Keywords: Aaronson-Ambainis conjecture, Quantum query complexity, Classical query complexity, Free probability, Completely bounded norm, Analysis of Boolean functions, Influence}
}
Document
On Randomized Reductions to the Random Strings
Abstract
We study the power of randomized polynomial-time non-adaptive reductions to the problem of approximating Kolmogorov complexity and its polynomial-time bounded variants.
As our first main result, we give a sharp dichotomy for randomized non-adaptive reducibility to approximating Kolmogorov complexity. We show that any computable language L that has a randomized polynomial-time non-adaptive reduction (satisfying a natural honesty condition) to ω(log(n))-approximating the Kolmogorov complexity is in AM ∩ coAM. On the other hand, using results of Hirahara [Shuichi Hirahara, 2020], it follows that every language in NEXP has a randomized polynomial-time non-adaptive reduction (satisfying the same honesty condition as before) to O(log(n))-approximating the Kolmogorov complexity.
As our second main result, we give the first negative evidence against the NP-hardness of polynomial-time bounded Kolmogorov complexity with respect to randomized reductions. We show that for every polynomial t', there is a polynomial t such that if there is a randomized time t' non-adaptive reduction (satisfying a natural honesty condition) from SAT to ω(log(n))-approximating K^t complexity, then either NE = coNE or 𝖤 has sub-exponential size non-deterministic circuits infinitely often.
Cite as
Michael Saks and Rahul Santhanam. On Randomized Reductions to the Random Strings. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 29:1-29:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{saks_et_al:LIPIcs.CCC.2022.29,
author = {Saks, Michael and Santhanam, Rahul},
title = {{On Randomized Reductions to the Random Strings}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {29:1--29:30},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.29},
URN = {urn:nbn:de:0030-drops-165912},
doi = {10.4230/LIPIcs.CCC.2022.29},
annote = {Keywords: Kolmogorov complexity, randomized reductions}
}
Document
Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes
Abstract
In this work, we initiate the study of proximity testing to Algebraic Geometry (AG) codes. An AG code C = C(𝒳, 𝒫, D) over an algebraic curve 𝒳 is a vector space associated to evaluations on 𝒫 ⊆ 𝒳 of functions in the Riemann-Roch space L_𝒳(D). The problem of testing proximity to an error-correcting code C consists in distinguishing between the case where an input word, given as an oracle, belongs to C and the one where it is far from every codeword of C. AG codes are good candidates to construct probabilistic proof systems, but there exists no efficient proximity tests for them. We aim to fill this gap.
We construct an Interactive Oracle Proof of Proximity (IOPP) for some families of AG codes by generalizing an IOPP for Reed-Solomon codes, known as the FRI protocol [Eli Ben-Sasson et al., 2018]. We identify suitable requirements for designing efficient IOPP systems for AG codes. Our approach relies on a neat decomposition of the Riemann-Roch space of any invariant divisor under a group action on a curve into several explicit Riemann-Roch spaces on the quotient curve. We provide sufficient conditions on an AG code C that allow to reduce a proximity testing problem for C to a membership problem for a significantly smaller code C'.
As concrete instantiations, we study AG codes on Kummer curves and curves in the Hermitian tower. The latter can be defined over polylogarithmic-size alphabet. We specialize the generic AG-IOPP construction to reach linear prover running time and logarithmic verification on Kummer curves, and quasilinear prover time with polylogarithmic verification on the Hermitian tower.
Cite as
Sarah Bordage, Mathieu Lhotel, Jade Nardi, and Hugues Randriam. Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 30:1-30:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bordage_et_al:LIPIcs.CCC.2022.30,
author = {Bordage, Sarah and Lhotel, Mathieu and Nardi, Jade and Randriam, Hugues},
title = {{Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {30:1--30:45},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.30},
URN = {urn:nbn:de:0030-drops-165923},
doi = {10.4230/LIPIcs.CCC.2022.30},
annote = {Keywords: Algebraic geometry codes, Interactive oracle proofs of proximity, Proximity testing}
}
Document
Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes
Abstract
We study the following natural question on random sets of points in 𝔽₂^m:
Given a random set of k points Z = {z₁, z₂, … , z_k} ⊆ 𝔽₂^m, what is the dimension of the space of degree at most r multilinear polynomials that vanish on all points in Z?
We show that, for r ≤ γ m (where γ > 0 is a small, absolute constant) and k = (1-ε)⋅binom(m, ≤ r) for any constant ε > 0, the space of degree at most r multilinear polynomials vanishing on a random set Z = {z_1,…, z_k} has dimension exactly binom(m, ≤ r) - k with probability 1 - o(1). This bound shows that random sets have a much smaller space of degree at most r multilinear polynomials vanishing on them, compared to the worst-case bound (due to Wei (IEEE Trans. Inform. Theory, 1991)) of binom(m, ≤ r) - binom(log₂ k, ≤ r) ≫ binom(m, ≤ r) - k.
Using this bound, we show that high-degree Reed-Muller codes (RM(m,d) with d > (1-γ) m) "achieve capacity" under the Binary Erasure Channel in the sense that, for any ε > 0, we can recover from (1-ε)⋅binom(m, ≤ m-d-1) random erasures with probability 1 - o(1). This also implies that RM(m,d) is also efficiently decodable from ≈ binom(m, ≤ m-(d/2)) random errors for the same range of parameters.
Cite as
Siddharth Bhandari, Prahladh Harsha, Ramprasad Saptharishi, and Srikanth Srinivasan. Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bhandari_et_al:LIPIcs.CCC.2022.31,
author = {Bhandari, Siddharth and Harsha, Prahladh and Saptharishi, Ramprasad and Srinivasan, Srikanth},
title = {{Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {31:1--31:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.31},
URN = {urn:nbn:de:0030-drops-165934},
doi = {10.4230/LIPIcs.CCC.2022.31},
annote = {Keywords: Reed-Muller codes, polynomials, weight-distribution, vanishing ideals, erasures, capacity}
}
Document
On the Partial Derivative Method Applied to Lopsided Set-Multilinear Polynomials
Abstract
We make progress on understanding a lower bound technique that was recently used by the authors to prove the first superpolynomial constant-depth circuit lower bounds against algebraic circuits.
More specifically, our previous work applied the well-known partial derivative method in a new setting, that of lopsided set-multilinear polynomials. A set-multilinear polynomial P ∈ 𝔽[X_1,…,X_d] (for disjoint sets of variables X_1,…,X_d) is a linear combination of monomials, each of which contains one variable from X_1,…,X_d. A lopsided space of set-multilinear polynomials is one where the sets X_1,…,X_d are allowed to have different sizes (we use the adjective "lopsided" to stress this feature). By choosing a suitable lopsided space of polynomials, and using a suitable version of the partial-derivative method for proving lower bounds, we were able to prove constant-depth superpolynomial set-multilinear formula lower bounds even for very low-degree polynomials (as long as d is a growing function of the number of variables N). This in turn implied lower bounds against general formulas of constant-depth.
A priori, there is nothing stopping these techniques from giving us lower bounds against algebraic formulas of any depth. We investigate the extent to which this lower bound can extend to greater depths. We prove the following results.
1) We observe that our choice of the lopsided space and the kind of partial-derivative method used can be modeled as the choice of a multiset W ⊆ [-1,1] of size d. Our first result completely characterizes, for any product-depth Δ, the best lower bound we can prove for set-multilinear formulas of product-depth Δ in terms of some combinatorial properties of W, that we call the depth-Δ tree bias of W.
2) We show that the maximum depth-3 tree bias, over multisets W of size d, is Θ(d^{1/4}). This shows a stronger formula lower bound of N^{Ω(d^{1/4})} for set-multilinear formulas of product-depth 3, and also puts a non-trivial constraint on the best lower bounds we can hope to prove at this depth in this framework (a priori, we could have hoped to prove a lower bound of N^{Ω(Δ d^{1/Δ})} at product-depth Δ).
3) Finally, we show that for small Δ, our proof technique cannot hope to prove lower bounds of the form N^{Ω(d^{1/poly(Δ)})}. This seems to strongly hint that new ideas will be required to prove lower bounds for formulas of unbounded depth.
Cite as
Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. On the Partial Derivative Method Applied to Lopsided Set-Multilinear Polynomials. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 32:1-32:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{limaye_et_al:LIPIcs.CCC.2022.32,
author = {Limaye, Nutan and Srinivasan, Srikanth and Tavenas, S\'{e}bastien},
title = {{On the Partial Derivative Method Applied to Lopsided Set-Multilinear Polynomials}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {32:1--32:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.32},
URN = {urn:nbn:de:0030-drops-165942},
doi = {10.4230/LIPIcs.CCC.2022.32},
annote = {Keywords: Partial Derivative Method, Barriers to Lower Bounds}
}
Document
Further Collapses in TFNP
Abstract
We show EOPL = PLS ∩ PPAD. Here the class EOPL consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubáček and Yogev (SICOMP 2020) and Fearnley et al. (JCSS 2020). In particular, our result yields a new simpler proof of the breakthrough collapse CLS = PLS ∩ PPAD by Fearnley et al. (STOC 2021). We also prove a companion result SOPL = PLS ∩ PPADS, where SOPL is the class associated with the Sink-of-Potential-Line problem.
Cite as
Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, and Ran Tao. Further Collapses in TFNP. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{goos_et_al:LIPIcs.CCC.2022.33,
author = {G\"{o}\"{o}s, Mika and Hollender, Alexandros and Jain, Siddhartha and Maystre, Gilbert and Pires, William and Robere, Robert and Tao, Ran},
title = {{Further Collapses in TFNP}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {33:1--33:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.33},
URN = {urn:nbn:de:0030-drops-165954},
doi = {10.4230/LIPIcs.CCC.2022.33},
annote = {Keywords: TFNP, PPAD, PLS, EOPL}
}
Document
Improved Pseudorandom Generators for AC⁰ Circuits
Abstract
We give PRG for depth-d, size-m AC⁰ circuits with seed length O(log^{d-1}(m)log(m/ε)log log(m)). Our PRG improves on previous work [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021] from various aspects. It has optimal dependence on 1/ε and is only one "log log(m)" away from the lower bound barrier. For the case of d = 2, the seed length tightly matches the best-known PRG for CNFs [Anindya De et al., 2010; Avishay Tal, 2017].
There are two technical ingredients behind our new result; both of them might be of independent interest. First, we use a partitioning-based approach to construct PRGs based on restriction lemmas for AC⁰. Previous works [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021] usually built PRGs on the Ajtai-Wigderson framework [Miklós Ajtai and Avi Wigderson, 1989]. Compared with them, the partitioning approach avoids the extra "log(n)" factor that usually arises from the Ajtai-Wigderson framework, allowing us to get the almost-tight seed length. The partitioning approach is quite general, and we believe it can help design PRGs for classes beyond constant-depth circuits.
Second, improving and extending [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021], we prove a full derandomization of the powerful multi-switching lemma [Johan Håstad, 2014]. We show that one can use a short random seed to sample a restriction, such that a family of DNFs simultaneously simplifies under the restriction with high probability. This answers an open question in [Zander Kelley, 2021]. Previous derandomizations were either partial (that is, they pseudorandomly choose variables to restrict, and then fix those variables to truly-random bits) or had sub-optimal seed length. In our application, having a fully-derandomized switching lemma is crucial, and the randomness-efficiency of our derandomization allows us to get an almost-tight seed length.
Cite as
Xin Lyu. Improved Pseudorandom Generators for AC⁰ Circuits. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 34:1-34:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{lyu:LIPIcs.CCC.2022.34,
author = {Lyu, Xin},
title = {{Improved Pseudorandom Generators for AC⁰ Circuits}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {34:1--34:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.34},
URN = {urn:nbn:de:0030-drops-165963},
doi = {10.4230/LIPIcs.CCC.2022.34},
annote = {Keywords: pseudorandom generators, derandomization, switching Lemmas, AC⁰}
}
Document
Characterizing Derandomization Through Hardness of Levin-Kolmogorov Complexity
Abstract
A central open problem in complexity theory concerns the question of whether all efficient randomized algorithms can be simulated by efficient deterministic algorithms. We consider this problem in the context of promise problems (i.e,. the prBPP v.s. prP problem) and show that for all sufficiently large constants c, the following are equivalent:
- prBPP = prP.
- For every BPTIME(n^c) algorithm M, and every sufficiently long z ∈ {0,1}ⁿ, there exists some x ∈ {0,1}ⁿ such that M fails to decide whether Kt(x∣z) is "very large" (≥ n-1) or "very small" (≤ O(log n)). where Kt(x∣z) denotes the Levin-Kolmogorov complexity of x conditioned on z. As far as we are aware, this yields the first full characterization of when prBPP = prP through the hardness of some class of problems. Previous hardness assumptions used for derandomization only provide a one-sided implication.
Cite as
Yanyi Liu and Rafael Pass. Characterizing Derandomization Through Hardness of Levin-Kolmogorov Complexity. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{liu_et_al:LIPIcs.CCC.2022.35,
author = {Liu, Yanyi and Pass, Rafael},
title = {{Characterizing Derandomization Through Hardness of Levin-Kolmogorov Complexity}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {35:1--35:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.35},
URN = {urn:nbn:de:0030-drops-165975},
doi = {10.4230/LIPIcs.CCC.2022.35},
annote = {Keywords: Derandomization, Kolmogorov Complexity, Hitting Set Generators}
}
Document
On One-Way Functions from NP-Complete Problems
Abstract
We present the first natural NP-complete problem whose average-case hardness w.r.t. the uniform distribution over instances is equivalent to the existence of one-way functions (OWFs). The problem, which originated in the 1960s, is the Conditional Time-Bounded Kolmogorov Complexity Problem: let K^t(x∣z) be the length of the shortest "program" that, given the "auxiliary input" z, outputs the string x within time t(|x|), and let McK^tP[ζ] be the set of strings (x,z,k) where |z| = ζ(|x|), |k| = log |x| and K^t(x∣z) < k, where, for our purposes, a "program" is defined as a RAM machine.
Our main result shows that for every polynomial t(n) ≥ n², there exists some polynomial ζ such that McK^tP[ζ] is NP-complete. We additionally extend the result of Liu-Pass (FOCS'20) to show that for every polynomial t(n) ≥ 1.1n, and every polynomial ζ(⋅), mild average-case hardness of McK^tP[ζ] is equivalent to the existence of OWFs. Taken together, these results provide the following crisp characterization of what is required to base OWFs on NP ⊈ BPP:
There exists concrete polynomials t,ζ such that "Basing OWFs on NP ⊈ BPP" is equivalent to providing a "worst-case to (mild) average-case reduction for McK^tP[ζ]".
In other words, the "holy-grail" of Cryptography (i.e., basing OWFs on NP ⊈ BPP) is equivalent to a basic question in algorithmic information theory.
As an independent contribution, we show that our NP-completeness result can be used to shed new light on the feasibility of the polynomial-time bounded symmetry of information assertion (Kolmogorov'68).
Cite as
Yanyi Liu and Rafael Pass. On One-Way Functions from NP-Complete Problems. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 36:1-36:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{liu_et_al:LIPIcs.CCC.2022.36,
author = {Liu, Yanyi and Pass, Rafael},
title = {{On One-Way Functions from NP-Complete Problems}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {36:1--36:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.36},
URN = {urn:nbn:de:0030-drops-165981},
doi = {10.4230/LIPIcs.CCC.2022.36},
annote = {Keywords: One-way Functions, NP-Completeness, Kolmogorov Complexity}
}
Document
Derandomization from Time-Space Tradeoffs
Abstract
A recurring challenge in the theory of pseudorandomness and circuit complexity is the explicit construction of "incompressible strings," i.e. finite objects which lack a specific type of structure or simplicity. In most cases, there is an associated NP search problem which we call the "compression problem," where we are given a candidate object and must either find a compressed/structured representation of it or determine that none exist. For a particular notion of compressibility, a natural question is whether an efficient algorithm for the compression problem would aide us in the construction of incompressible objects. Consider the following two instances of this question:
1) Does an efficient algorithm for circuit minimization imply efficient constructions of hard truth tables?
2) Does an efficient algorithm for factoring integers imply efficient constructions of large prime numbers? In this work, we connect these kinds of questions to the long-standing challenge of proving time-space tradeoffs for Turing machines, and proving stronger separations between the RAM and 1-tape computation models. In particular, one of our main theorems shows that modest time-space tradeoffs for deterministic exponential time, or separations between basic Turing machine memory models, would imply a positive answer to both (1) and (2). These results apply to the derandomization of a wider class of explicit construction problems, where we have some efficient compression scheme that encodes n-bit strings using < n bits, and we aim to construct an n-bit string which cannot be recovered from its encoding.
Cite as
Oliver Korten. Derandomization from Time-Space Tradeoffs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 37:1-37:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{korten:LIPIcs.CCC.2022.37,
author = {Korten, Oliver},
title = {{Derandomization from Time-Space Tradeoffs}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {37:1--37:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.37},
URN = {urn:nbn:de:0030-drops-165993},
doi = {10.4230/LIPIcs.CCC.2022.37},
annote = {Keywords: Pseudorandomness, circuit complexity, total functions}
}
Document
Improved Low-Depth Set-Multilinear Circuit Lower Bounds
Abstract
In this paper, we prove strengthened lower bounds for constant-depth set-multilinear formulas. More precisely, we show that over any field, there is an explicit polynomial f in VNP defined over n² variables, and of degree n, such that any product-depth Δ set-multilinear formula computing f has size at least n^Ω(n^{1/Δ}/Δ). The hard polynomial f comes from the class of Nisan-Wigderson (NW) design-based polynomials.
Our lower bounds improve upon the recent work of Limaye, Srinivasan and Tavenas (STOC 2022), where a lower bound of the form (log n)^Ω(Δ n^{1/Δ}) was shown for the size of product-depth Δ set-multilinear formulas computing the iterated matrix multiplication (IMM) polynomial of the same degree and over the same number of variables as f. Moreover, our lower bounds are novel for any Δ ≥ 2.
The precise quantitative expression in our lower bound is interesting also because the lower bounds we obtain are "sharp" in the sense that any asymptotic improvement would imply general set-multilinear circuit lower bounds via depth reduction results.
In the setting of general set-multilinear formulas, a lower bound of the form n^Ω(log n) was already obtained by Raz (J. ACM 2009) for the more general model of multilinear formulas. The techniques of LST (which extend the techniques of the same authors in (FOCS 2021)) give a different route to set-multilinear formula lower bounds, and allow them to obtain a lower bound of the form (log n)^Ω(log n) for the size of general set-multilinear formulas computing the IMM polynomial. Our proof techniques are another variation on those of LST, and enable us to show an improved lower bound (matching that of Raz) of the form n^Ω(log n), albeit for the same polynomial f in VNP (the NW polynomial). As observed by LST, if the same n^Ω(log n) size lower bounds for unbounded-depth set-multilinear formulas could be obtained for the IMM polynomial, then using the self-reducibility of IMM and using hardness escalation results, this would imply super-polynomial lower bounds for general algebraic formulas.
Cite as
Deepanshu Kush and Shubhangi Saraf. Improved Low-Depth Set-Multilinear Circuit Lower Bounds. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kush_et_al:LIPIcs.CCC.2022.38,
author = {Kush, Deepanshu and Saraf, Shubhangi},
title = {{Improved Low-Depth Set-Multilinear Circuit Lower Bounds}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {38:1--38:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-241-9},
ISSN = {1868-8969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.38},
URN = {urn:nbn:de:0030-drops-166003},
doi = {10.4230/LIPIcs.CCC.2022.38},
annote = {Keywords: algebraic circuit complexity, complexity measure, set-multilinear formulas}
}