Volume
LIPIcs, Volume 300
39th Computational Complexity Conference (CCC 2024)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Computational Complexity Conference (CCC)
Event
CCC 2024, July 22-25, 2024, Ann Arbor, MI, USAEditor
Publication Details
- published at: 2024-07-15
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-331-7
- DBLP: db/conf/coco/coco2024
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Complete Volume
LIPIcs, Volume 300, CCC 2024, Complete Volume
Abstract
LIPIcs, Volume 300, CCC 2024, Complete Volume
Cite as
39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 1-952, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@Proceedings{santhanam:LIPIcs.CCC.2024,
title = {{LIPIcs, Volume 300, CCC 2024, Complete Volume}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {1--952},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024},
URN = {urn:nbn:de:0030-drops-203955},
doi = {10.4230/LIPIcs.CCC.2024},
annote = {Keywords: LIPIcs, Volume 300, CCC 2024, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 0:i-0:xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{santhanam:LIPIcs.CCC.2024.0,
author = {Santhanam, Rahul},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {0:i--0:xiv},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.0},
URN = {urn:nbn:de:0030-drops-203961},
doi = {10.4230/LIPIcs.CCC.2024.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
A Technique for Hardness Amplification Against AC⁰
Abstract
We study hardness amplification in the context of two well-known "moderate" average-case hardness results for AC⁰ circuits. First, we investigate the extent to which AC⁰ circuits of depth d can approximate AC⁰ circuits of some larger depth d + k. The case k = 1 is resolved by Håstad, Rossman, Servedio, and Tan’s celebrated average-case depth hierarchy theorem (JACM 2017). Our contribution is a significantly stronger correlation bound when k ≥ 3. Specifically, we show that there exists a linear-size AC⁰_{d + k} circuit h : {0, 1}ⁿ → {0, 1} such that for every AC⁰_d circuit g, either g has size exp(n^{Ω(1/d)}), or else g agrees with h on at most a (1/2 + ε)-fraction of inputs where ε = exp(-(1/d) ⋅ Ω(log n)^{k-1}). For comparison, Håstad, Rossman, Servedio, and Tan’s result has ε = n^{-Θ(1/d)}. Second, we consider the majority function. It is well known that the majority function is moderately hard for AC⁰ circuits (and stronger classes). Our contribution is a stronger correlation bound for the XOR of t copies of the n-bit majority function, denoted MAJ_n^{⊕ t}. We show that if g is an AC⁰_d circuit of size S, then g agrees with MAJ_n^{⊕ t} on at most a (1/2 + ε)-fraction of inputs, where ε = (O(log S)^{d - 1} / √n)^t.
To prove these results, we develop a hardness amplification technique that is tailored to a specific type of circuit lower bound proof. In particular, one way to show that a function h is moderately hard for AC⁰ circuits is to (a) design some distribution over random restrictions or random projections, (b) show that AC⁰ circuits simplify to shallow decision trees under these restrictions/projections, and finally (c) show that after applying the restriction/projection, h is moderately hard for shallow decision trees with respect to an appropriate distribution. We show that (roughly speaking) if h can be proven to be moderately hard by a proof with that structure, then XORing multiple copies of h amplifies its hardness. Our analysis involves a new kind of XOR lemma for decision trees, which might be of independent interest.
Cite as
William M. Hoza. A Technique for Hardness Amplification Against AC⁰. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hoza:LIPIcs.CCC.2024.1,
author = {Hoza, William M.},
title = {{A Technique for Hardness Amplification Against AC⁰}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {1:1--1:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.1},
URN = {urn:nbn:de:0030-drops-203977},
doi = {10.4230/LIPIcs.CCC.2024.1},
annote = {Keywords: Bounded-depth circuits, average-case lower bounds, hardness amplification, XOR lemmas}
}
Document
Streaming Zero-Knowledge Proofs
Abstract
Streaming interactive proofs (SIPs) enable a space-bounded algorithm with one-pass access to a massive stream of data to verify a computation that requires large space, by communicating with a powerful but untrusted prover.
This work initiates the study of zero-knowledge proofs for data streams. We define the notion of zero-knowledge in the streaming setting and construct zero-knowledge SIPs for the two main algorithmic building blocks in the streaming interactive proofs literature: the sumcheck and polynomial evaluation protocols. To the best of our knowledge all known streaming interactive proofs are based on either of these tools, and indeed, this allows us to obtain zero-knowledge SIPs for central streaming problems such as index, point and range queries, median, frequency moments, and inner product.
Our protocols are efficient in terms of time and space, as well as communication: the verifier algorithm’s space complexity is polylog(n) and, after a non-interactive setup that uses a random string of near-linear length, the remaining parameters are n^o(1).
En route, we develop an algorithmic toolkit for designing zero-knowledge data stream protocols, consisting of an algebraic streaming commitment protocol and a temporal commitment protocol. Our analyses rely on delicate algebraic and information-theoretic arguments and reductions from average-case communication complexity.
Cite as
Graham Cormode, Marcel Dall'Agnol, Tom Gur, and Chris Hickey. Streaming Zero-Knowledge Proofs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 2:1-2:66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{cormode_et_al:LIPIcs.CCC.2024.2,
author = {Cormode, Graham and Dall'Agnol, Marcel and Gur, Tom and Hickey, Chris},
title = {{Streaming Zero-Knowledge Proofs}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {2:1--2:66},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.2},
URN = {urn:nbn:de:0030-drops-203988},
doi = {10.4230/LIPIcs.CCC.2024.2},
annote = {Keywords: Zero-knowledge proofs, streaming algorithms, computational complexity}
}
Document
Solving Unique Games over Globally Hypercontractive Graphs
Abstract
We study the complexity of affine Unique-Games (UG) over globally hypercontractive graphs, which are graphs that are not small set expanders but admit a useful and succinct characterization of all small sets that violate the small-set expansion property. This class of graphs includes the Johnson and Grassmann graphs, which have played a pivotal role in recent PCP constructions for UG, and their generalizations via high-dimensional expanders.
We show new rounding techniques for higher degree sum-of-squares (SoS) relaxations for worst-case optimization. In particular, our algorithm shows how to round "low-entropy" pseudodistributions, broadly extending the algorithmic framework of [Mitali Bafna et al., 2021]. At a high level, [Mitali Bafna et al., 2021] showed how to round pseudodistributions for problems where there is a "unique" good solution. We extend their framework by exhibiting a rounding for problems where there might be "few good solutions".
Our result suggests that UG is easy on globally hypercontractive graphs, and therefore highlights the importance of graphs that lack such a characterization in the context of PCP reductions for UG.
Cite as
Mitali Bafna and Dor Minzer. Solving Unique Games over Globally Hypercontractive Graphs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bafna_et_al:LIPIcs.CCC.2024.3,
author = {Bafna, Mitali and Minzer, Dor},
title = {{Solving Unique Games over Globally Hypercontractive Graphs}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {3:1--3:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.3},
URN = {urn:nbn:de:0030-drops-203996},
doi = {10.4230/LIPIcs.CCC.2024.3},
annote = {Keywords: unique games, approximation algorithms}
}
Document
Derandomizing Logspace with a Small Shared Hard Drive
Abstract
We obtain new catalytic algorithms for space-bounded derandomization. In the catalytic computation model introduced by (Buhrman, Cleve, Koucký, Loff, and Speelman STOC 2013), we are given a small worktape, and a larger catalytic tape that has an arbitrary initial configuration. We may edit this tape, but it must be exactly restored to its initial configuration at the completion of the computation. We prove that BPSPACE[S] ⊆ CSPACE[S,S²] where BPSPACE[S] corresponds to randomized space S computation, and CSPACE[S,C] corresponds to catalytic algorithms that use O(S) bits of workspace and O(C) bits of catalytic space. Previously, only BPSPACE[S] ⊆ CSPACE[S,2^O(S)] was known. In fact, we prove a general tradeoff, that for every α ∈ [1,1.5], BPSPACE[S] ⊆ CSPACE[S^α,S^(3-α)]. We do not use the algebraic techniques of prior work on catalytic computation. Instead, we develop an algorithm that branches based on if the catalytic tape is conditionally random, and instantiate this primitive in a recursive framework. Our result gives an alternate proof of the best known time-space tradeoff for BPSPACE[S], due to (Cai, Chakaravarthy, and van Melkebeek, Theory Comput. Sys. 2006). As a final application, we extend our results to solve search problems in CSPACE[S,S²]. As far as we are aware, this constitutes the first study of search problems in the catalytic computing model.
Cite as
Edward Pyne. Derandomizing Logspace with a Small Shared Hard Drive. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{pyne:LIPIcs.CCC.2024.4,
author = {Pyne, Edward},
title = {{Derandomizing Logspace with a Small Shared Hard Drive}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {4:1--4:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.4},
URN = {urn:nbn:de:0030-drops-204006},
doi = {10.4230/LIPIcs.CCC.2024.4},
annote = {Keywords: Catalytic computation, space-bounded computation, derandomization}
}
Document
Explicit Time and Space Efficient Encoders Exist Only with Random Access
Abstract
We give the first explicit constant rate, constant relative distance, linear codes with an encoder that runs in time n^{1 + o(1)} and space polylog(n) provided random access to the message. Prior to this work, the only such codes were non-explicit, for instance repeat accumulate codes [Divsalar et al., 1998] and the codes described in [Gál et al., 2013]. To construct our codes, we also give explicit, efficiently invertible, lossless condensers with constant entropy gap and polylogarithmic seed length.
In contrast to encoders with random access to the message, we show that encoders with sequential access to the message can not run in almost linear time and polylogarithmic space. Our notion of sequential access is much stronger than streaming access.
Cite as
Joshua Cook and Dana Moshkovitz. Explicit Time and Space Efficient Encoders Exist Only with Random Access. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 5:1-5:54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{cook_et_al:LIPIcs.CCC.2024.5,
author = {Cook, Joshua and Moshkovitz, Dana},
title = {{Explicit Time and Space Efficient Encoders Exist Only with Random Access}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {5:1--5:54},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.5},
URN = {urn:nbn:de:0030-drops-204015},
doi = {10.4230/LIPIcs.CCC.2024.5},
annote = {Keywords: Time-Space Trade Offs, Error Correcting Codes, Encoders, Explicit Constructions, Streaming Lower Bounds, Sequential Access, Time-Space Lower Bounds, Lossless Condensers, Invertible Condensers, Condensers}
}
Document
The Entangled Quantum Polynomial Hierarchy Collapses
Abstract
We introduce the entangled quantum polynomial hierarchy, QEPH, as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove QEPH collapses to its second level. In fact, we show that a polynomial number of alternations collapses to just two. As a consequence, QEPH = QRG(1), the class of problems having one-turn quantum refereed games, which is known to be contained in PSPACE. This is in contrast to the unentangled quantum polynomial hierarchy, QPH, which contains QMA(2).
We also introduce DistributionQCPH, a generalization of the quantum-classical polynomial hierarchy QCPH where the provers send probability distributions over strings (instead of strings). We prove DistributionQCPH = QCPH, suggesting that only quantum superposition (not classical probability) increases the computational power of these hierarchies. To prove this equality, we generalize a game-theoretic result of Lipton and Young (1994) which says that, without loss of generality, the provers can send uniform distributions over a polynomial-size support. We also prove the analogous result for the polynomial hierarchy, i.e., DistributionPH = PH.
Finally, we show that PH and QCPH are contained in QPH, resolving an open question of Gharibian et al. (2022).
Cite as
Sabee Grewal and Justin Yirka. The Entangled Quantum Polynomial Hierarchy Collapses. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{grewal_et_al:LIPIcs.CCC.2024.6,
author = {Grewal, Sabee and Yirka, Justin},
title = {{The Entangled Quantum Polynomial Hierarchy Collapses}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {6:1--6:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.6},
URN = {urn:nbn:de:0030-drops-204028},
doi = {10.4230/LIPIcs.CCC.2024.6},
annote = {Keywords: Polynomial hierarchy, Entangled proofs, Correlated proofs, Minimax}
}
Document
Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy
Abstract
The following question arises naturally in the study of graph streaming algorithms:
Is there any graph problem which is "not too hard", in that it can be solved efficiently with total communication (nearly) linear in the number n of vertices, and for which, nonetheless, any streaming algorithm with Õ(n) space (i.e., a semi-streaming algorithm) needs a polynomial n^Ω(1) number of passes?
Assadi, Chen, and Khanna [STOC 2019] were the first to prove that this is indeed the case. However, the lower bounds that they obtained are for rather non-standard graph problems.
Our first main contribution is to present the first polynomial-pass lower bounds for natural "not too hard" graph problems studied previously in the streaming model: k-cores and degeneracy. We devise a novel communication protocol for both problems with near-linear communication, thus showing that k-cores and degeneracy are natural examples of "not too hard" problems. Indeed, previous work have developed single-pass semi-streaming algorithms for approximating these problems. In contrast, we prove that any semi-streaming algorithm for exactly solving these problems requires (almost) Ω(n^{1/3}) passes.
The lower bound follows by a reduction from a generalization of the hidden pointer chasing (HPC) problem of Assadi, Chen, and Khanna, which is also the basis of their earlier semi-streaming lower bounds.
Our second main contribution is improved round-communication lower bounds for the underlying communication problems at the basis of these reductions:
- We improve the previous lower bound of Assadi, Chen, and Khanna for HPC to achieve optimal bounds for this problem.
- We further observe that all current reductions from HPC can also work with a generalized version of this problem that we call MultiHPC, and prove an even stronger and optimal lower bound for this generalization. These two results collectively allow us to improve the resulting pass lower bounds for semi-streaming algorithms by a polynomial factor, namely, from n^{1/5} to n^{1/3} passes.
Cite as
Sepehr Assadi, Prantar Ghosh, Bruno Loff, Parth Mittal, and Sagnik Mukhopadhyay. Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{assadi_et_al:LIPIcs.CCC.2024.7,
author = {Assadi, Sepehr and Ghosh, Prantar and Loff, Bruno and Mittal, Parth and Mukhopadhyay, Sagnik},
title = {{Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {7:1--7:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.7},
URN = {urn:nbn:de:0030-drops-204035},
doi = {10.4230/LIPIcs.CCC.2024.7},
annote = {Keywords: Graph streaming, Lower bounds, Communication complexity, k-Cores and degeneracy}
}
Document
Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries
Abstract
Recently, Kumar and Mon reached a significant milestone by constructing asymptotically good relaxed locally correctable codes (RLCCs) with poly-logarithmic query complexity. Specifically, they constructed n-bit RLCCs with O(log^{69} n) queries. Their construction relies on a clever reduction to locally testable codes (LTCs), capitalizing on recent breakthrough works in LTCs. As for lower bounds, Gur and Lachish (SICOMP 2021) proved that any asymptotically-good RLCC must make Ω̃(√{log n}) queries. Hence emerges the intriguing question regarding the identity of the least value 1/2 ≤ e ≤ 69 for which asymptotically-good RLCCs with query complexity (log n)^{e+o(1)} exist.
In this work, we make substantial progress in narrowing the gap by devising asymptotically-good RLCCs with a query complexity of (log n)^{2+o(1)}. The key insight driving our work lies in recognizing that the strong guarantee of local testability overshoots the requirements for the Kumar-Mon reduction. In particular, we prove that we can replace the LTCs by "vanilla" expander codes which indeed have the necessary property: local testability in the code’s vicinity.
Cite as
Gil Cohen and Tal Yankovitz. Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{cohen_et_al:LIPIcs.CCC.2024.8,
author = {Cohen, Gil and Yankovitz, Tal},
title = {{Asymptotically-Good RLCCs with (log n)^(2+o(1)) Queries}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {8:1--8:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.8},
URN = {urn:nbn:de:0030-drops-204045},
doi = {10.4230/LIPIcs.CCC.2024.8},
annote = {Keywords: Relaxed locally decodable codes, Relxaed locally correctable codes, RLCC, RLDC}
}
Document
Lifting Dichotomies
Abstract
Lifting theorems are used for transferring lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure A for some function f, we compose f with a carefully chosen gadget function g and get essentially the same lower bound on a complexity measure B for the lifted function f ⋄ g. Lifting theorems have a number of applications in many different areas such as circuit complexity, communication complexity, proof complexity, etc. One of the main question in the context of lifting is how to choose a suitable gadget g. Generally, to get better results, i.e., to minimize the losses when transferring lower bounds, we need the gadget to be of a constant size (number of inputs). Unfortunately, in many settings we know lifting results only for gadgets of size that grows with the size of f, and it is unclear whether it can be improved to a constant size gadget. This motivates us to identify the properties of gadgets that make lifting possible.
In this paper, we systematically study the question "For which gadgets does the lifting result hold?" in the following four settings: lifting from decision tree depth to decision tree size, lifting from conjunction DAG width to conjunction DAG size, lifting from decision tree depth to parity decision tree depth and size, and lifting from block sensitivity to deterministic and randomized communication complexities. In all the cases, we prove the complete classification of gadgets by exposing the properties of gadgets that make lifting results hold. The structure of the results shows that there is no intermediate cases - for every gadget there is either a polynomial lifting or no lifting at all. As a byproduct of our studies, we prove the log-rank conjecture for the class of functions that can be represented as f ⋄ OR ⋄ XOR for some function f.
In this extended abstract, the proofs are omitted. Full proofs are given in the full version [Yaroslav Alekseev et al., 2024].
Cite as
Yaroslav Alekseev, Yuval Filmus, and Alexander Smal. Lifting Dichotomies. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{alekseev_et_al:LIPIcs.CCC.2024.9,
author = {Alekseev, Yaroslav and Filmus, Yuval and Smal, Alexander},
title = {{Lifting Dichotomies}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {9:1--9:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.9},
URN = {urn:nbn:de:0030-drops-204051},
doi = {10.4230/LIPIcs.CCC.2024.9},
annote = {Keywords: decision trees, log-rank conjecture, lifting, parity decision trees}
}
Document
Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs
Abstract
Affine extractors give some of the best-known lower bounds for various computational models, such as AC⁰ circuits, parity decision trees, and general Boolean circuits. However, they are not known to give strong lower bounds for read-once branching programs (ROBPs). In a recent work, Gryaznov, Pudlák, and Talebanfard (CCC' 22) introduced a stronger version of affine extractors known as directional affine extractors, together with a generalization of ROBPs where each node can make linear queries, and showed that the former implies strong lower bound for a certain type of the latter known as strongly read-once linear branching programs (SROLBPs). Their main result gives explicit constructions of directional affine extractors for entropy k > 2n/3, which implies average-case complexity 2^{n/3-o(n)} against SROLBPs with exponentially small correlation. A follow-up work by Chattopadhyay and Liao (CCC' 23) improves the hardness to 2^{n-o(n)} at the price of increasing the correlation to polynomially large, via a new connection to sumset extractors introduced by Chattopadhyay and Li (STOC' 16) and explicit constructions of such extractors by Chattopadhyay and Liao (STOC' 22). Both works left open the questions of better constructions of directional affine extractors and improved average-case complexity against SROLBPs in the regime of small correlation.
This paper provides a much more in-depth study of directional affine extractors, SROLBPs, and ROBPs. Our main results include:
- An explicit construction of directional affine extractors with k = o(n) and exponentially small error, which gives average-case complexity 2^{n-o(n)} against SROLBPs with exponentially small correlation, thus answering the two open questions raised in previous works.
- An explicit function in AC⁰ that gives average-case complexity 2^{(1-δ)n} against ROBPs with negligible correlation, for any constant δ > 0. Previously, no such average-case hardness is known, and the best size lower bound for any function in AC⁰ against ROBPs is 2^Ω(n).
One of the key ingredients in our constructions is a new linear somewhere condenser for affine sources, which is based on dimension expanders. The condenser also leads to an unconditional improvement of the entropy requirement of explicit affine extractors with negligible error. We further show that the condenser also works for general weak random sources, under the Polynomial Freiman-Ruzsa Theorem in 𝖥₂ⁿ, recently proved by Gowers, Green, Manners, and Tao (arXiv' 23).
Cite as
Xin Li and Yan Zhong. Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{li_et_al:LIPIcs.CCC.2024.10,
author = {Li, Xin and Zhong, Yan},
title = {{Explicit Directional Affine Extractors and Improved Hardness for Linear Branching Programs}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {10:1--10:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.10},
URN = {urn:nbn:de:0030-drops-204060},
doi = {10.4230/LIPIcs.CCC.2024.10},
annote = {Keywords: Randomness Extractors, Affine, Read-once Linear Branching Programs, Low-degree polynomials, AC⁰ circuits}
}
Document
Linear-Size Boolean Circuits for Multiselection
Abstract
We study the circuit complexity of the multiselection problem: given an input string x ∈ {0,1}ⁿ along with indices i_1,… ,i_q ∈ [n], output (x_{i_1},… ,x_{i_q}). A trivial lower bound for the circuit size is the input length n + q⋅log(n), but the straightforward construction has size Θ(q⋅n).
Our main result is an O(n+q⋅log³(n))-size and O(log(n+q))-depth circuit for multiselection. In particular, for any q ≤ n/log³(n) the circuit has linear size and logarithmic depth. Prior to our work no linear-size circuit for multiselection was known for any q = ω(1) and regardless of depth.
Cite as
Justin Holmgren and Ron Rothblum. Linear-Size Boolean Circuits for Multiselection. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{holmgren_et_al:LIPIcs.CCC.2024.11,
author = {Holmgren, Justin and Rothblum, Ron},
title = {{Linear-Size Boolean Circuits for Multiselection}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {11:1--11:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.11},
URN = {urn:nbn:de:0030-drops-204070},
doi = {10.4230/LIPIcs.CCC.2024.11},
annote = {Keywords: Private Information Retrieval, Batch Selection, Boolean Circuits}
}
Document
A Subquadratic Upper Bound on Sum-Of-Squares Composition Formulas
Abstract
For every n, we construct a sum-of-squares identity (∑_{i=1}^n x_i²) (∑_{j=1}^n y_j²) = ∑_{k=1}^s f_k², where f_k are bilinear forms with complex coefficients and s = O(n^1.62). Previously, such a construction was known with s = O(n²/log n). The same bound holds over any field of positive characteristic.
Cite as
Pavel Hrubeš. A Subquadratic Upper Bound on Sum-Of-Squares Composition Formulas. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 12:1-12:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hrubes:LIPIcs.CCC.2024.12,
author = {Hrube\v{s}, Pavel},
title = {{A Subquadratic Upper Bound on Sum-Of-Squares Composition Formulas}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {12:1--12:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.12},
URN = {urn:nbn:de:0030-drops-204082},
doi = {10.4230/LIPIcs.CCC.2024.12},
annote = {Keywords: Sum-of-squares composition formulas, Hurwitz’s problem, non-commutative arithmetic circuit}
}
Document
Hard Submatrices for Non-Negative Rank and Communication Complexity
Abstract
Given a non-negative real matrix M of non-negative rank at least r, can we witness this fact by a small submatrix of M? While Moitra (SIAM J. Comput. 2013) proved that this cannot be achieved exactly, we show that such a witnessing is possible approximately: an m×n matrix of non-negative rank r always contains a submatrix with at most r³ rows and columns with non-negative rank at least Ω(r/(log n log m)). A similar result is proved for the 1-partition number of a Boolean matrix and, consequently, also for its two-player deterministic communication complexity. Tightness of the latter estimate is closely related to the log-rank conjecture of Lovász and Saks.
Cite as
Pavel Hrubeš. Hard Submatrices for Non-Negative Rank and Communication Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 13:1-13:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hrubes:LIPIcs.CCC.2024.13,
author = {Hrube\v{s}, Pavel},
title = {{Hard Submatrices for Non-Negative Rank and Communication Complexity}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {13:1--13:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.13},
URN = {urn:nbn:de:0030-drops-204097},
doi = {10.4230/LIPIcs.CCC.2024.13},
annote = {Keywords: Non-negative rank, communication complexity, extension complexity}
}
Document
Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture
Abstract
When a group acts on a set, it naturally partitions it into orbits, giving rise to orbit problems. These are natural algorithmic problems, as symmetries are central in numerous questions and structures in physics, mathematics, computer science, optimization, and more. Accordingly, it is of high interest to understand their computational complexity. Recently, Bürgisser et al. (2021) gave the first polynomial-time algorithms for orbit problems of torus actions, that is, actions of commutative continuous groups on Euclidean space. In this work, motivated by theoretical and practical applications, we study the computational complexity of robust generalizations of these orbit problems, which amount to approximating the distance of orbits in ℂⁿ up to a factor γ ≥ 1. In particular, this allows deciding whether two inputs are approximately in the same orbit or far from being so. On the one hand, we prove the NP-hardness of this problem for γ = n^Ω(1/log log n) by reducing the closest vector problem for lattices to it. On the other hand, we describe algorithms for solving this problem for an approximation factor γ = exp(poly(n)). Our algorithms combine tools from invariant theory and algorithmic lattice theory, and they also provide group elements witnessing the proximity of the given orbits (in contrast to the algebraic algorithms of prior work). We prove that they run in polynomial time if and only if a version of the famous number-theoretic abc-conjecture holds - establishing a new and surprising connection between computational complexity and number theory.
Cite as
Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson. Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 14:1-14:48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{burgisser_et_al:LIPIcs.CCC.2024.14,
author = {B\"{u}rgisser, Peter and Do\u{g}an, Mahmut Levent and Makam, Visu and Walter, Michael and Wigderson, Avi},
title = {{Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {14:1--14:48},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.14},
URN = {urn:nbn:de:0030-drops-204100},
doi = {10.4230/LIPIcs.CCC.2024.14},
annote = {Keywords: computational invariant theory, geometric complexity theory, orbit problems, abc-conjecture, closest vector problem}
}
Document
Quantum Automating TC⁰-Frege Is LWE-Hard
Abstract
We prove the first hardness results against efficient proof search by quantum algorithms. We show that under Learning with Errors (LWE), the standard lattice-based cryptographic assumption, no quantum algorithm can weakly automate TC⁰-Frege. This extends the line of results of Krajíček and Pudlák (Information and Computation, 1998), Bonet, Pitassi, and Raz (FOCS, 1997), and Bonet, Domingo, Gavaldà, Maciel, and Pitassi (Computational Complexity, 2004), who showed that Extended Frege, TC⁰-Frege and AC⁰-Frege, respectively, cannot be weakly automated by classical algorithms if either the RSA cryptosystem or the Diffie-Hellman key exchange protocol are secure. To the best of our knowledge, this is the first interaction between quantum computation and propositional proof search.
Cite as
Noel Arteche, Gaia Carenini, and Matthew Gray. Quantum Automating TC⁰-Frege Is LWE-Hard. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 15:1-15:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{arteche_et_al:LIPIcs.CCC.2024.15,
author = {Arteche, Noel and Carenini, Gaia and Gray, Matthew},
title = {{Quantum Automating TC⁰-Frege Is LWE-Hard}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {15:1--15:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.15},
URN = {urn:nbn:de:0030-drops-204117},
doi = {10.4230/LIPIcs.CCC.2024.15},
annote = {Keywords: automatability, post-quantum cryptography, feasible interpolation}
}
Document
A Strong Direct Sum Theorem for Distributional Query Complexity
Abstract
Consider the expected query complexity of computing the k-fold direct product f^{⊗ k} of a function f to error ε with respect to a distribution μ^k. One strategy is to sequentially compute each of the k copies to error ε/k with respect to μ and apply the union bound. We prove a strong direct sum theorem showing that this naive strategy is essentially optimal. In particular, computing a direct product necessitates a blowup in both query complexity and error.
Strong direct sum theorems contrast with results that only show a blowup in query complexity or error but not both. There has been a long line of such results for distributional query complexity, dating back to (Impagliazzo, Raz, Wigderson 1994) and (Nisan, Rudich, Saks 1994), but a strong direct sum theorem that holds for all functions in the standard query model had been elusive.
A key idea in our work is the first use of the Hardcore Theorem (Impagliazzo 1995) in the context of query complexity. We prove a new resilience lemma that accompanies it, showing that the hardcore of f^{⊗k} is likely to remain dense under arbitrary partitions of the input space.
Cite as
Guy Blanc, Caleb Koch, Carmen Strassle, and Li-Yang Tan. A Strong Direct Sum Theorem for Distributional Query Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 16:1-16:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{blanc_et_al:LIPIcs.CCC.2024.16,
author = {Blanc, Guy and Koch, Caleb and Strassle, Carmen and Tan, Li-Yang},
title = {{A Strong Direct Sum Theorem for Distributional Query Complexity}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {16:1--16:30},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.16},
URN = {urn:nbn:de:0030-drops-204123},
doi = {10.4230/LIPIcs.CCC.2024.16},
annote = {Keywords: Query complexity, direct product theorem, hardcore theorem}
}
Document
Local Enumeration and Majority Lower Bounds
Abstract
Depth-3 circuit lower bounds and k-SAT algorithms are intimately related; the state-of-the-art Σ^k_3-circuit lower bound (Or-And-Or circuits with bottom fan-in at most k) and the k-SAT algorithm of Paturi, Pudlák, Saks, and Zane (J. ACM'05) are based on the same combinatorial theorem regarding k-CNFs. In this paper we define a problem which reveals new interactions between the two, and suggests a concrete approach to significantly stronger circuit lower bounds and improved k-SAT algorithms. For a natural number k and a parameter t, we consider the Enum(k, t) problem defined as follows: given an n-variable k-CNF and an initial assignment α, output all satisfying assignments at Hamming distance t(n) of α, assuming that there are no satisfying assignments of Hamming distance less than t(n) of α. We observe that an upper bound b(n, k, t) on the complexity of Enum(k, t) simultaneously implies depth-3 circuit lower bounds and k-SAT algorithms:
- Depth-3 circuits: Any Σ^k_3 circuit computing the Majority function has size at least binom(n,n/2)/b(n, k, n/2).
- k-SAT: There exists an algorithm solving k-SAT in time O(∑_{t=1}^{n/2}b(n, k, t)).
A simple construction shows that b(n, k, n/2) ≥ 2^{(1 - O(log(k)/k))n}. Thus, matching upper bounds for b(n, k, n/2) would imply a Σ^k_3-circuit lower bound of 2^Ω(log(k)n/k) and a k-SAT upper bound of 2^{(1 - Ω(log(k)/k))n}. The former yields an unrestricted depth-3 lower bound of 2^ω(√n) solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis.
In this paper, we propose a randomized algorithm for Enum(k, t) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum(3, n/2). We show that the expected running time of our algorithm is 1.598ⁿ, substantially improving on the trivial bound of 3^{n/2} ≃ 1.732ⁿ. This already improves Σ^3_3 lower bounds for Majority function to 1.251ⁿ. The previous bound was 1.154ⁿ which follows from the work of Håstad, Jukna, and Pudlák (Comput. Complex.'95).
By restricting ourselves to monotone CNFs, Enum(k, t) immediately becomes a hypergraph Turán problem. Therefore our techniques might be of independent interest in extremal combinatorics.
Cite as
Mohit Gurumukhani, Ramamohan Paturi, Pavel Pudlák, Michael Saks, and Navid Talebanfard. Local Enumeration and Majority Lower Bounds. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 17:1-17:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{gurumukhani_et_al:LIPIcs.CCC.2024.17,
author = {Gurumukhani, Mohit and Paturi, Ramamohan and Pudl\'{a}k, Pavel and Saks, Michael and Talebanfard, Navid},
title = {{Local Enumeration and Majority Lower Bounds}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {17:1--17:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.17},
URN = {urn:nbn:de:0030-drops-204136},
doi = {10.4230/LIPIcs.CCC.2024.17},
annote = {Keywords: Depth 3 circuits, k-CNF satisfiability, Circuit lower bounds, Majority function}
}
Document
Pseudorandomness, Symmetry, Smoothing: I
Abstract
We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that
- achieve an optimal lower bound on their statistical distance to any bounded-uniform distribution. This closes a line of research initiated by Alon, Goldreich, and Mansour in 2003, and improves on a result by O'Donnell and Zhao.
- have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke.
- rule out a popular paradigm for constructing pseudorandom generators, originating in a 1989 work by Ajtai and Wigderson. This again answers a question raised by several researchers. For branching programs, our result matches a bound by Forbes and Kelley. Our small-bias distributions above are symmetric. We show that the xor of any two symmetric small-bias distributions fools any bounded function. Hence our examples cannot be extended to the xor of two small-bias distributions, another popular paradigm whose power remains unknown. We also generalize and simplify the proof of a result of Bazzi.
Cite as
Harm Derksen, Peter Ivanov, Chin Ho Lee, and Emanuele Viola. Pseudorandomness, Symmetry, Smoothing: I. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 18:1-18:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{derksen_et_al:LIPIcs.CCC.2024.18,
author = {Derksen, Harm and Ivanov, Peter and Lee, Chin Ho and Viola, Emanuele},
title = {{Pseudorandomness, Symmetry, Smoothing: I}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {18:1--18:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.18},
URN = {urn:nbn:de:0030-drops-204144},
doi = {10.4230/LIPIcs.CCC.2024.18},
annote = {Keywords: pseudorandomness, k-wise uniform distributions, small-bias distributions, noise, symmetric tests, thresholds, Krawtchouk polynomials}
}
Document
Information Dissemination via Broadcasts in the Presence of Adversarial Noise
Abstract
We initiate the study of error correcting codes over the multi-party adversarial broadcast channel. Specifically, we consider the classic information dissemination problem where n parties, each holding an input bit, wish to know each other’s input. For this, they communicate in rounds, where, in each round, one designated party sends a bit to all other parties over a channel governed by an adversary that may corrupt a constant fraction of the received communication. We mention that the dissemination problem was studied in the stochastic noise model since the 80’s.
While stochastic noise in multi-party channels has received quite a bit of attention, the case of adversarial noise has largely been avoided, as such channels cannot handle more than a 1/n-fraction of errors. Indeed, this many errors allow an adversary to completely corrupt the incoming or outgoing communication for one of the parties and fail the protocol. Curiously, we show that by eliminating these "trivial" attacks, one can get a simple protocol resilient to a constant fraction of errors. Thus, a model that rules out such attacks is both necessary and sufficient to get a resilient protocol.
The main shortcoming of our dissemination protocol is its length: it requires Θ(n²) communication rounds whereas n rounds suffice in the absence of noise. Our main result is a matching lower bound of Ω(n²) on the length of any dissemination protocol in our model. Our proof first "gets rid" of the channel noise by converting it to a form of "input noise", showing that a noisy dissemination protocol implies a (noiseless) protocol for a version of the direct sum gap-majority problem. We conclude the proof with a tight lower bound for the latter problem, which may be of independent interest.
Cite as
Klim Efremenko, Gillat Kol, Dmitry Paramonov, Ran Raz, and Raghuvansh R. Saxena. Information Dissemination via Broadcasts in the Presence of Adversarial Noise. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 19:1-19:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{efremenko_et_al:LIPIcs.CCC.2024.19,
author = {Efremenko, Klim and Kol, Gillat and Paramonov, Dmitry and Raz, Ran and Saxena, Raghuvansh R.},
title = {{Information Dissemination via Broadcasts in the Presence of Adversarial Noise}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {19:1--19:33},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.19},
URN = {urn:nbn:de:0030-drops-204159},
doi = {10.4230/LIPIcs.CCC.2024.19},
annote = {Keywords: Radio Networks, Interactive Coding, Error Correcting Codes}
}
Document
Lower Bounds for Set-Multilinear Branching Programs
Abstract
In this paper, we prove super-polynomial lower bounds for the model of sum of ordered set-multilinear algebraic branching programs, each with a possibly different ordering (∑smABP). Specifically, we give an explicit nd-variate polynomial of degree d such that any ∑smABP computing it must have size n^ω(1) for d as low as ω(log n). Notably, this constitutes the first such lower bound in the low degree regime. Moreover, for d = poly(n), we demonstrate an exponential lower bound. This result generalizes the seminal work of Nisan (STOC, 1991), which proved an exponential lower bound for a single ordered set-multilinear ABP.
The significance of our lower bounds is underscored by the recent work of Bhargav, Dwivedi, and Saxena (TAMC, 2024), which showed that super-polynomial lower bounds against a sum of ordered set-multilinear branching programs - for a polynomial of sufficiently low degree - would imply super-polynomial lower bounds against general ABPs, thereby resolving Valiant’s longstanding conjecture that the permanent polynomial can not be computed efficiently by ABPs. More precisely, their work shows that if one could obtain such lower bounds when the degree is bounded by O(log n/ log log n), then it would imply super-polynomial lower bounds against general ABPs.
Our results strengthen the works of Arvind & Raja (Chic. J. Theor. Comput. Sci., 2016) and Bhargav, Dwivedi & Saxena (TAMC, 2024), as well as the works of Ramya & Rao (Theor. Comput. Sci., 2020) and Ghoshal & Rao (International Computer Science Symposium in Russia, 2021), each of which established lower bounds for related or restricted versions of this model. They also strongly answer a question from the former two, which asked to prove super-polynomial lower bounds for general ∑smABP.
Cite as
Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, and Amir Shpilka. Lower Bounds for Set-Multilinear Branching Programs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2024.20,
author = {Chatterjee, Prerona and Kush, Deepanshu and Saraf, Shubhangi and Shpilka, Amir},
title = {{Lower Bounds for Set-Multilinear Branching Programs}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {20:1--20:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.20},
URN = {urn:nbn:de:0030-drops-204167},
doi = {10.4230/LIPIcs.CCC.2024.20},
annote = {Keywords: Lower Bounds, Algebraic Branching Programs, Set-multilinear polynomials}
}
Document
Public-Key Pseudoentanglement and the Hardness of Learning Ground State Entanglement Structure
Abstract
Given a local Hamiltonian, how difficult is it to determine the entanglement structure of its ground state? We show that this problem is computationally intractable even if one is only trying to decide if the ground state is volume-law vs near area-law entangled. We prove this by constructing strong forms of pseudoentanglement in a public-key setting, where the circuits used to prepare the states are public knowledge. In particular, we construct two families of quantum circuits which produce volume-law vs near area-law entangled states, but nonetheless the classical descriptions of the circuits are indistinguishable under the Learning with Errors (LWE) assumption. Indistinguishability of the circuits then allows us to translate our construction to Hamiltonians. Our work opens new directions in Hamiltonian complexity, for example whether it is difficult to learn certain phases of matter.
Cite as
Adam Bouland, Bill Fefferman, Soumik Ghosh, Tony Metger, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou. Public-Key Pseudoentanglement and the Hardness of Learning Ground State Entanglement Structure. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 21:1-21:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bouland_et_al:LIPIcs.CCC.2024.21,
author = {Bouland, Adam and Fefferman, Bill and Ghosh, Soumik and Metger, Tony and Vazirani, Umesh and Zhang, Chenyi and Zhou, Zixin},
title = {{Public-Key Pseudoentanglement and the Hardness of Learning Ground State Entanglement Structure}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {21:1--21:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.21},
URN = {urn:nbn:de:0030-drops-204175},
doi = {10.4230/LIPIcs.CCC.2024.21},
annote = {Keywords: Quantum computing, Quantum complexity theory, entanglement}
}
Document
Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP
Abstract
We show that for any integer d > 0, depth-d Frege systems are NP-hard to automate. Namely, given a set S of depth-d formulas, it is NP-hard to find a depth-d Frege refutation of S in time polynomial in the size of the shortest such refutation. This extends the result of Atserias and Müller [JACM, 2020] for the non-automatability of resolution - a depth-1 Frege system - to Frege systems of any depth d > 0.
Cite as
Theodoros Papamakarios. Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{papamakarios:LIPIcs.CCC.2024.22,
author = {Papamakarios, Theodoros},
title = {{Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {22:1--22:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.22},
URN = {urn:nbn:de:0030-drops-204187},
doi = {10.4230/LIPIcs.CCC.2024.22},
annote = {Keywords: Proof complexity, Automatability, Bounded-depth Frege}
}
Document
Exponential Separation Between Powers of Regular and General Resolution over Parities
Abstract
Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities is an outstanding problem that has received a lot of attention after its introduction by Itsykson and Sokolov [Dmitry Itsykson and Dmitry Sokolov, 2014]. Very recently, Efremenko, Garlík and Itsykson [Klim Efremenko et al., 2023] proved the first exponential lower bounds on the size of ResLin proofs that were additionally restricted to be bottom-regular. We show that there are formulas for which such regular ResLin proofs of unsatisfiability continue to have exponential size even though there exist short proofs of their unsatisfiability in ordinary, non-regular resolution. This is the first super-polynomial separation between the power of general ResLin and that of regular ResLin for any natural notion of regularity.
Our argument, while building upon the work of Efremenko et al. [Klim Efremenko et al., 2023], uses additional ideas from the literature on lifting theorems.
Cite as
Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák. Exponential Separation Between Powers of Regular and General Resolution over Parities. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 23:1-23:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bhattacharya_et_al:LIPIcs.CCC.2024.23,
author = {Bhattacharya, Sreejata Kishor and Chattopadhyay, Arkadev and Dvo\v{r}\'{a}k, Pavel},
title = {{Exponential Separation Between Powers of Regular and General Resolution over Parities}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {23:1--23:32},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.23},
URN = {urn:nbn:de:0030-drops-204191},
doi = {10.4230/LIPIcs.CCC.2024.23},
annote = {Keywords: Proof Complexity, Regular Reslin, Branching Programs, Lifting}
}
Document
Distribution-Free Proofs of Proximity
Abstract
Motivated by the fact that input distributions are often unknown in advance, distribution-free property testing considers a setting in which the algorithmic task is to accept functions f: [n] → {0,1} having a certain property Π and reject functions that are ε-far from Π, where the distance is measured according to an arbitrary and unknown input distribution 𝒟 ∼ [n]. As usual in property testing, the tester is required to do so while making only a sublinear number of input queries, but as the distribution is unknown, we also allow a sublinear number of samples from the distribution 𝒟.
In this work we initiate the study of distribution-free interactive proofs of proximity (df-IPPs) in which the distribution-free testing algorithm is assisted by an all powerful but untrusted prover. Our main result is that for any problem Π ∈ NC, any proximity parameter ε > 0, and any (trade-off) parameter τ ≤ √n, we construct a df-IPP for Π with respect to ε, that has query and sample complexities τ+O(1/ε), and communication complexity Õ(n/τ + 1/ε). For τ as above and sufficiently large ε (namely, when ε > τ/n), this result matches the parameters of the best-known general purpose IPPs in the standard uniform setting. Moreover, for such τ, its parameters are optimal up to poly-logarithmic factors under reasonable cryptographic assumptions for the same regime of ε as the uniform setting, i.e., when ε ≥ 1/τ.
For smaller values of ε (i.e., when ε < τ/n), our protocol has communication complexity Ω(1/ε), which is worse than the Õ(n/τ) communication complexity of the uniform IPPs (with the same query complexity). With the aim of improving on this gap, we further show that for IPPs over specialised, but large distribution families, such as sufficiently smooth distributions and product distributions, the communication complexity can be reduced to Õ(n/τ^{1-o(1)}). In addition, we show that for certain natural families of languages, such as symmetric and (relaxed) self-correctable languages, it is possible to further improve the efficiency of distribution-free IPPs.
Cite as
Hugo Aaronson, Tom Gur, Ninad Rajgopal, and Ron D. Rothblum. Distribution-Free Proofs of Proximity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{aaronson_et_al:LIPIcs.CCC.2024.24,
author = {Aaronson, Hugo and Gur, Tom and Rajgopal, Ninad and Rothblum, Ron D.},
title = {{Distribution-Free Proofs of Proximity}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {24:1--24:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.24},
URN = {urn:nbn:de:0030-drops-204204},
doi = {10.4230/LIPIcs.CCC.2024.24},
annote = {Keywords: Property Testing, Interactive Proofs, Distribution-Free Property Testing}
}
Document
On the Degree of Polynomials Computing Square Roots Mod p
Abstract
For an odd prime p, we say f(X) ∈ F_p[X] computes square roots in F_p if, for all nonzero perfect squares a ∈ F_p, we have f(a)² = a.
When p ≡ 3 mod 4, it is well known that f(X) = X^{(p+1)/4} computes square roots. This degree is surprisingly low (and in fact lowest possible), since we have specified (p-1)/2 evaluations (up to sign) of the polynomial f(X). On the other hand, for p ≡ 1 mod 4 there was previously no nontrivial bound known on the lowest degree of a polynomial computing square roots in F_p.
We show that for all p ≡ 1 mod 4, the degree of a polynomial computing square roots has degree at least p/3. Our main new ingredient is a general lemma which may be of independent interest: powers of a low degree polynomial cannot have too many consecutive zero coefficients. The proof method also yields a robust version: any polynomial that computes square roots for 99% of the squares also has degree almost p/3.
In the other direction, Agou, Deliglése, and Nicolas [Agou et al., 2003] showed that for infinitely many p ≡ 1 mod 4, the degree of a polynomial computing square roots can be as small as 3p/8.
Cite as
Kiran S. Kedlaya and Swastik Kopparty. On the Degree of Polynomials Computing Square Roots Mod p. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{kedlaya_et_al:LIPIcs.CCC.2024.25,
author = {Kedlaya, Kiran S. and Kopparty, Swastik},
title = {{On the Degree of Polynomials Computing Square Roots Mod p}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {25:1--25:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.25},
URN = {urn:nbn:de:0030-drops-204219},
doi = {10.4230/LIPIcs.CCC.2024.25},
annote = {Keywords: Algebraic Computation, Polynomials, Computing Square roots, Reed-Solomon Codes}
}
Document
Dimension Independent Disentanglers from Unentanglement and Applications
Abstract
Quantum entanglement, a distinctive form of quantum correlation, has become a key enabling ingredient in diverse applications in quantum computation, complexity, cryptography, etc. However, the presence of unwanted adversarial entanglement also poses challenges and even prevents the correct behaviour of many protocols and applications.
In this paper, we explore methods to "break" the quantum correlations. Specifically, we construct a dimension-independent k-partite disentangler (like) channel from bipartite unentangled input. In particular, we show: For every d,𝓁 ≥ k ∈ ℕ^+, there is an efficient channel Λ : ℂ^{d𝓁} ⊗ ℂ^{d𝓁} → ℂ^{dk} such that for every bipartite separable density operator ρ₁⊗ ρ₂, the output Λ(ρ₁⊗ρ₂) is close to a k-partite separable state. Concretely, for some distribution μ on states from C^d, ║ Λ(ρ₁⊗ρ₂) - ∫ |ψ⟩⟨ψ|^{⊗k} dμ(ψ) ║₁ ≤ Õ((k³/𝓁)^{1/4}).
Moreover, Λ(|ψ⟩⟨ψ|^{⊗𝓁} ⊗ |ψ⟩⟨ψ|^{⊗𝓁}) = |ψ⟩⟨ψ|^{⊗k}. Without the bipartite unentanglement assumption, the above bound is conjectured to be impossible and would imply QMA(2) = QMA.
Leveraging multipartite unentanglement ensured by our disentanglers, we achieve the following: (i) a new proof that QMA(2) admits arbitrary gap amplification; (ii) a variant of the swap test and product test with improved soundness, addressing a major limitation of their original versions. More importantly, we demonstrate that unentangled quantum proofs of almost general real amplitudes capture NEXP, thereby greatly relaxing the non-negative amplitudes assumption in the recent work of QMA^+(2) = NEXP [Jeronimo and Wu, STOC 2023]. Specifically, our findings show that to capture NEXP, it suffices to have unentangled proofs of the form |ψ⟩ = √a |ψ_{+}⟩ + √{1-a} |ψ_{-}⟩ where |ψ_{+}⟩ has non-negative amplitudes, |ψ_{-}⟩ only has negative amplitudes and |a-(1-a)| ≥ 1/poly(n) with a ∈ [0,1]. Additionally, we present a protocol achieving an almost largest possible completeness-soundness gap before obtaining QMA^ℝ(k) = NEXP, namely, a 1/poly(n) additive improvement to the gap results in this equality.
Cite as
Fernando Granha Jeronimo and Pei Wu. Dimension Independent Disentanglers from Unentanglement and Applications. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 26:1-26:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{jeronimo_et_al:LIPIcs.CCC.2024.26,
author = {Jeronimo, Fernando Granha and Wu, Pei},
title = {{Dimension Independent Disentanglers from Unentanglement and Applications}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {26:1--26:28},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.26},
URN = {urn:nbn:de:0030-drops-204228},
doi = {10.4230/LIPIcs.CCC.2024.26},
annote = {Keywords: QMA(2), disentangler, quantum proofs}
}
Document
Baby PIH: Parameterized Inapproximability of Min CSP
Abstract
The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP theorem in the world of parameterized complexity. It asserts that no FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only (1-ε)-satisfiable (where the parameter is the number of variables) for some constant 0 < ε < 1.
We consider a minimization version of CSPs (Min-CSP), where one may assign r values to each variable, and the goal is to ensure that every constraint is satisfied by some choice among the r × r pairs of values assigned to its variables (call such a CSP instance r-list-satisfiable). We prove the following strong parameterized inapproximability for Min CSP: For every r ≥ 1, it is W[1]-hard to tell if a 2CSP instance is satisfiable or is not even r-list-satisfiable. We refer to this statement as "Baby PIH", following the recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the combinatorial arguments underlying the Baby PCP theorem, overcoming some basic obstacles that arise in the parameterized setting. Furthermore, our reduction runs in time polynomially bounded in both the number of variables and the alphabet size, and thus implies the Baby PCP theorem as well.
Cite as
Venkatesan Guruswami, Xuandi Ren, and Sai Sandeep. Baby PIH: Parameterized Inapproximability of Min CSP. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{guruswami_et_al:LIPIcs.CCC.2024.27,
author = {Guruswami, Venkatesan and Ren, Xuandi and Sandeep, Sai},
title = {{Baby PIH: Parameterized Inapproximability of Min CSP}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {27:1--27:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.27},
URN = {urn:nbn:de:0030-drops-204237},
doi = {10.4230/LIPIcs.CCC.2024.27},
annote = {Keywords: Parameterized Inapproximability Hypothesis, Constraint Satisfaction Problems}
}
Document
Finding Missing Items Requires Strong Forms of Randomness
Abstract
Adversarially robust streaming algorithms are required to process a stream of elements and produce correct outputs, even when each stream element can be chosen as a function of earlier algorithm outputs. As with classic streaming algorithms, which must only be correct for the worst-case fixed stream, adversarially robust algorithms with access to randomness can use significantly less space than deterministic algorithms. We prove that for the Missing Item Finding problem in streaming, the space complexity also significantly depends on how adversarially robust algorithms are permitted to use randomness. (In contrast, the space complexity of classic streaming algorithms does not depend as strongly on the way randomness is used.)
For Missing Item Finding on streams of length 𝓁 with elements in {1,…,n}, and ≤ 1/poly(𝓁) error, we show that when 𝓁 = O(2^√{log n}), "random seed" adversarially robust algorithms, which only use randomness at initialization, require 𝓁^Ω(1) bits of space, while "random tape" adversarially robust algorithms, which may make random decisions at any time, may use O(polylog(𝓁)) random bits. When 𝓁 is between n^Ω(1) and O(√n), "random tape" adversarially robust algorithms need 𝓁^Ω(1) space, while "random oracle" adversarially robust algorithms, which can read from a long random string for free, may use O(polylog(𝓁)) space. The space lower bound for the "random seed" case follows, by a reduction given in prior work, from a lower bound for pseudo-deterministic streaming algorithms given in this paper.
Cite as
Amit Chakrabarti and Manuel Stoeckl. Finding Missing Items Requires Strong Forms of Randomness. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chakrabarti_et_al:LIPIcs.CCC.2024.28,
author = {Chakrabarti, Amit and Stoeckl, Manuel},
title = {{Finding Missing Items Requires Strong Forms of Randomness}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {28:1--28:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.28},
URN = {urn:nbn:de:0030-drops-204242},
doi = {10.4230/LIPIcs.CCC.2024.28},
annote = {Keywords: Data streaming, lower bounds, space complexity, adversarial robustness, derandomization, sketching, sampling}
}
Document
Exact Search-To-Decision Reductions for Time-Bounded Kolmogorov Complexity
Abstract
A search-to-decision reduction is a procedure that allows one to find a solution to a problem from the mere ability to decide when a solution exists. The existence of a search-to-decision reduction for time-bounded Kolmogorov complexity, i.e., the problem of checking if a string x can be generated by a t-time bounded program of description length s, is a long-standing open problem that dates back to the 1960s.
In this work, we obtain new average-case and worst-case search-to-decision reductions for the complexity measure 𝖪^t and its randomized analogue rK^t:
1) (Conditional Errorless and Error-Prone Reductions for 𝖪^t) Under the assumption that 𝖤 requires exponential size circuits, we design polynomial-time average-case search-to-decision reductions for 𝖪^t in both errorless and error-prone settings. In fact, under the easiness of deciding 𝖪^t under the uniform distribution, we obtain a search algorithm for any given polynomial-time samplable distribution. In the error-prone reduction, the search algorithm works in the more general setting of conditional 𝖪^t complexity, i.e., it finds a minimum length t-time bound program for generating x given a string y.
2) (Unconditional Errorless Reduction for rK^t) We obtain an unconditional polynomial-time average-case search-to-decision reduction for rK^t in the errorless setting. Similarly to the results described above, we obtain a search algorithm for each polynomial-time samplable distribution, assuming the existence of a decision algorithm under the uniform distribution. To our knowledge, this is the first unconditional sub-exponential time search-to-decision reduction among the measures 𝖪^t and rK^t that works with respect to any given polynomial-time samplable distribution.
3) (Worst-Case to Average-Case Reductions) Under the errorless average-case easiness of deciding rK^t, we design a worst-case search algorithm running in time 2^O(n/log n) that produces a minimum length randomized t-time program for every input string x ∈ {0,1}ⁿ, with the caveat that it only succeeds on some explicitly computed sub-exponential time bound t ≤ 2^{n^ε} that depends on x. A similar result holds for 𝖪^t, under the assumption that 𝖤 requires exponential size circuits. In these results, the corresponding search problem is solved exactly, i.e., a successful run of the search algorithm outputs a t-time bounded program for x of minimum length, as opposed to an approximately optimal program of slightly larger description length or running time.
Cite as
Shuichi Hirahara, Valentine Kabanets, Zhenjian Lu, and Igor C. Oliveira. Exact Search-To-Decision Reductions for Time-Bounded Kolmogorov Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 29:1-29:56, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hirahara_et_al:LIPIcs.CCC.2024.29,
author = {Hirahara, Shuichi and Kabanets, Valentine and Lu, Zhenjian and Oliveira, Igor C.},
title = {{Exact Search-To-Decision Reductions for Time-Bounded Kolmogorov Complexity}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {29:1--29:56},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.29},
URN = {urn:nbn:de:0030-drops-204256},
doi = {10.4230/LIPIcs.CCC.2024.29},
annote = {Keywords: average-case complexity, Kolmogorov complexity, search-to-decision reductions}
}
Document
The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise
Abstract
The class MIP^* of quantum multiprover interactive proof systems with entanglement is much more powerful than its classical counterpart MIP [Babai et al., 1991; Zhengfeng Ji et al., 2020; Zhengfeng Ji et al., 2020]: while MIP = NEXP, the quantum class MIP^* is equal to RE, a class including the halting problem. This is because the provers in MIP^* can share unbounded quantum entanglement. However, recent works [Qin and Yao, 2021; Qin and Yao, 2023] have shown that this advantage is significantly reduced if the provers' shared state contains noise. This paper attempts to exactly characterize the effect of noise on the computational power of quantum multiprover interactive proof systems. We investigate the quantum two-prover one-round interactive system MIP^*[poly,O(1)], where the verifier sends polynomially many bits to the provers and the provers send back constantly many bits. We show noise completely destroys the computational advantage given by shared entanglement in this model. Specifically, we show that if the provers are allowed to share arbitrarily many EPR states, where each EPR state is affected by an arbitrarily small constant amount of noise, the resulting complexity class is equivalent to NEXP = MIP. This improves significantly on the previous best-known bound of NEEEXP (nondeterministic triply exponential time) [Qin and Yao, 2021]. We also show that this collapse in power is due to the noise, rather than the O(1) answer size, by showing that allowing for noiseless EPR states gives the class the full power of RE = MIP^*[poly, poly]. Along the way, we develop two technical tools of independent interest. First, we give a new, deterministic tester for the positivity of an exponentially large matrix, provided it has a low-degree Fourier decomposition in terms of Pauli matrices. Secondly, we develop a new invariance principle for smooth matrix functions having bounded third-order Fréchet derivatives or which are Lipschitz continuous.
Cite as
Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao. The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 30:1-30:71, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{dong_et_al:LIPIcs.CCC.2024.30,
author = {Dong, Yangjing and Fu, Honghao and Natarajan, Anand and Qin, Minglong and Xu, Haochen and Yao, Penghui},
title = {{The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {30:1--30:71},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.30},
URN = {urn:nbn:de:0030-drops-204263},
doi = {10.4230/LIPIcs.CCC.2024.30},
annote = {Keywords: Interactive proofs, Quantum complexity theory, Quantum entanglement, Fourier analysis, Matrix analysis, Invariance principle, Derandomization, PCP, Locally testable code, Positivity testing}
}
Document
Low-Depth Algebraic Circuit Lower Bounds over Any Field
Abstract
The recent breakthrough of Limaye, Srinivasan and Tavenas [Limaye et al., 2022] (LST) gave the first super-polynomial lower bounds against low-depth algebraic circuits, for any field of zero (or sufficiently large) characteristic. It was an open question to extend this result to small-characteristic ([Limaye et al., 2022; Govindasamy et al., 2022; Fournier et al., 2023]), which in particular is relevant for an approach to prove superpolynomial AC⁰[p]-Frege lower bounds ([Govindasamy et al., 2022]).
In this work, we prove super-polynomial algebraic circuit lower bounds against low-depth algebraic circuits over any field, with the same parameters as LST (or even matching the improved parameters of Bhargav, Dutta, and Saxena [Bhargav et al., 2022]). We give two proofs. The first is logical, showing that even though the proof of LST naively fails in small characteristic, the proof is sufficiently algebraic that generic transfer results imply the result over characteristic zero implies the result over all fields. Motivated by this indirect proof, we then proceed to give a second constructive proof, replacing the field-dependent set-multilinearization result of LST with a set-multilinearization that works over any field, by using the Binet-Minc identity [Minc, 1979].
Cite as
Michael A. Forbes. Low-Depth Algebraic Circuit Lower Bounds over Any Field. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{forbes:LIPIcs.CCC.2024.31,
author = {Forbes, Michael A.},
title = {{Low-Depth Algebraic Circuit Lower Bounds over Any Field}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {31:1--31:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.31},
URN = {urn:nbn:de:0030-drops-204271},
doi = {10.4230/LIPIcs.CCC.2024.31},
annote = {Keywords: algebraic circuits, lower bounds, low-depth circuits, positive characteristic}
}
Document
BPL ⊆ L-AC¹
Abstract
Whether BPL = 𝖫 (which is conjectured to be equal) or even whether BPL ⊆ NL, is a big open problem in theoretical computer science. It is well known that 𝖫 ⊆ NL ⊆ L-AC¹. In this work we show that BPL ⊆ L-AC¹ also holds. Our proof is based on a new iteration method for boosting precision in approximating matrix powering, which is inspired by the Richardson Iteration method developed in a recent line of work [AmirMahdi Ahmadinejad et al., 2020; Edward Pyne and Salil P. Vadhan, 2021; Gil Cohen et al., 2021; William M. Hoza, 2021; Gil Cohen et al., 2023; Aaron (Louie) Putterman and Edward Pyne, 2023; Lijie Chen et al., 2023]. We also improve the algorithm for approximate counting in low-depth L-AC circuits from an additive error setting to a multiplicative error setting.
Cite as
Kuan Cheng and Yichuan Wang. BPL ⊆ L-AC¹. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{cheng_et_al:LIPIcs.CCC.2024.32,
author = {Cheng, Kuan and Wang, Yichuan},
title = {{BPL ⊆ L-AC¹}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {32:1--32:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.32},
URN = {urn:nbn:de:0030-drops-204282},
doi = {10.4230/LIPIcs.CCC.2024.32},
annote = {Keywords: Randomized Space Complexity, Circuit Complexity, Derandomization}
}
Document
Failure of Feasible Disjunction Property for k-DNF Resolution and NP-Hardness of Automating It
Abstract
We show that for every integer k ≥ 2, the Res(k) propositional proof system does not have the weak feasible disjunction property. Next, we generalize a result of Atserias and Müller [Atserias and Müller, 2019] to Res(k). We show that if NP is not included in P (resp. QP, SUBEXP) then for every integer k ≥ 1, Res(k) is not automatable in polynomial (resp. quasi-polynomial, subexponential) time.
Cite as
Michal Garlík. Failure of Feasible Disjunction Property for k-DNF Resolution and NP-Hardness of Automating It. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 33:1-33:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{garlik:LIPIcs.CCC.2024.33,
author = {Garl{\'\i}k, Michal},
title = {{Failure of Feasible Disjunction Property for k-DNF Resolution and NP-Hardness of Automating It}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {33:1--33:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.33},
URN = {urn:nbn:de:0030-drops-204295},
doi = {10.4230/LIPIcs.CCC.2024.33},
annote = {Keywords: reflection principle, feasible disjunction property, k-DNF Resolution}
}
Document
Search-To-Decision Reductions for Kolmogorov Complexity
Abstract
A long-standing open problem dating back to the 1960s is whether there exists a search-to-decision reduction for the time-bounded Kolmogorov complexity problem - that is, the problem of determining whether the length of the shortest time-t program generating a given string x is at most s.
In this work, we consider the more "robust" version of the time-bounded Kolmogorov complexity problem, referred to as the GapMINKT problem, where given a size bound s and a running time bound t, the goal is to determine whether there exists a poly(t,|x|)-time program of length s+O(log |x|) that generates x. We present the first non-trivial search-to-decision reduction R for the GapMINKT problem; R has a running-time bound of 2^{ε n} for any ε > 0 and additionally only queries its oracle on "thresholds" s of size s+O(log |x|). As such, we get that any algorithm with running-time (resp. circuit size) 2^{α s} poly(|x|,t,s) for solving GapMINKT (given an instance (x,t,s), yields an algorithm for finding a witness with running-time (resp. circuit size) 2^{(α+ε) s} poly(|x|,t,s).
Our second result is a polynomial-time search-to-decision reduction for the time-bounded Kolmogorov complexity problem in the average-case regime. Such a reduction was recently shown by Liu and Pass (FOCS'20), heavily relying on cryptographic techniques. Our reduction is more direct and additionally has the advantage of being length-preserving, and as such also applies in the exponential time/size regime.
A central component in both of these results is the use of Kolmogorov and Levin’s Symmetry of Information Theorem.
Cite as
Noam Mazor and Rafael Pass. Search-To-Decision Reductions for Kolmogorov Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 34:1-34:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{mazor_et_al:LIPIcs.CCC.2024.34,
author = {Mazor, Noam and Pass, Rafael},
title = {{Search-To-Decision Reductions for Kolmogorov Complexity}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {34:1--34:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.34},
URN = {urn:nbn:de:0030-drops-204308},
doi = {10.4230/LIPIcs.CCC.2024.34},
annote = {Keywords: Kolmogorov complexity, search to decision}
}
Document
Finer-Grained Hardness of Kernel Density Estimation
Abstract
In batch Kernel Density Estimation (KDE) for a kernel function f : ℝ^m × ℝ^m → ℝ, we are given as input 2n points x^{(1)}, …, x^{(n)}, y^{(1)}, …, y^{(n)} ∈ ℝ^m in dimension m, as well as a vector v ∈ ℝⁿ. These inputs implicitly define the n × n kernel matrix K given by K[i,j] = f(x^{(i)}, y^{(j)}). The goal is to compute a vector v ∈ ℝⁿ which approximates K w, i.e., with || Kw - v||_∞ < ε ||w||₁. For illustrative purposes, consider the Gaussian kernel f(x,y) : = e^{-||x-y||₂²}. The classic approach to this problem is the famous Fast Multipole Method (FMM), which runs in time n ⋅ O(log^m(ε^{-1})) and is particularly effective in low dimensions because of its exponential dependence on m. Recently, as the higher-dimensional case m ≥ Ω(log n) has seen more applications in machine learning and statistics, new algorithms have focused on this setting: an algorithm using discrepancy theory, which runs in time O(n / ε), and an algorithm based on the polynomial method, which achieves inverse polynomial accuracy in almost linear time when the input points have bounded square diameter B < o(log n).
A recent line of work has proved fine-grained lower bounds, with the goal of showing that the "curse of dimensionality" arising in FMM is necessary assuming the Strong Exponential Time Hypothesis (SETH). Backurs et al. [NeurIPS 2017] first showed the hardness of a variety of Empirical Risk Minimization problems including KDE for Gaussian-like kernels in the case with high dimension m = Ω(log n) and large scale B = Ω(log n). Alman et al. [FOCS 2020] later developed new reductions in roughly this same parameter regime, leading to lower bounds for more general kernels, but only for very small error ε < 2^{- log^{Ω(1)} (n)}.
In this paper, we refine the approach of Alman et al. to show new lower bounds in all parameter regimes, closing gaps between the known algorithms and lower bounds. For example:
- In the setting where m = Clog n and B = o(log n), we prove Gaussian KDE requires n^{2-o(1)} time to achieve additive error ε < Ω(m/B)^{-m}, matching the performance of the polynomial method up to low-order terms.
- In the low dimensional setting m = o(log n), we show that Gaussian KDE requires n^{2-o(1)} time to achieve ε such that log log (ε^{-1}) > ̃ Ω ((log n)/m), matching the error bound achievable by FMM up to low-order terms. To our knowledge, no nontrivial lower bound was previously known in this regime. Our approach also generalizes to any parameter regime and any kernel. For example, we achieve similar fine-grained hardness results for any kernel with slowly-decaying Taylor coefficients such as the Cauchy kernel. Our new lower bounds make use of an intricate analysis of the "counting matrix", a special case of the kernel matrix focused on carefully-chosen evaluation points. As a key technical lemma, we give a novel approach to bounding the entries of its inverse by using Schur polynomials from algebraic combinatorics.
Cite as
Josh Alman and Yunfeng Guan. Finer-Grained Hardness of Kernel Density Estimation. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 35:1-35:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{alman_et_al:LIPIcs.CCC.2024.35,
author = {Alman, Josh and Guan, Yunfeng},
title = {{Finer-Grained Hardness of Kernel Density Estimation}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {35:1--35:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.35},
URN = {urn:nbn:de:0030-drops-204311},
doi = {10.4230/LIPIcs.CCC.2024.35},
annote = {Keywords: Kernel Density Estimation, Fine-Grained Complexity, Schur Polynomials}
}
Document
Gap MCSP Is Not (Levin) NP-Complete in Obfustopia
Abstract
We demonstrate that under believable cryptographic hardness assumptions, Gap versions of standard meta-complexity problems, such as the Minimum Circuit Size Problem (MCSP) and the Minimum Time-Bounded Kolmogorov Complexity problem (MKTP) are not NP-complete w.r.t. Levin (i.e., witness-preserving many-to-one) reductions. In more detail:
- Assuming the existence of indistinguishability obfuscation, and subexponentially-secure one-way functions, an appropriate Gap version of MCSP is not NP-complete under randomized Levin-reductions.
- Assuming the existence of subexponentially-secure indistinguishability obfuscation, subexponentially-secure one-way functions and injective PRGs, an appropriate Gap version of MKTP is not NP-complete under randomized Levin-reductions.
Cite as
Noam Mazor and Rafael Pass. Gap MCSP Is Not (Levin) NP-Complete in Obfustopia. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{mazor_et_al:LIPIcs.CCC.2024.36,
author = {Mazor, Noam and Pass, Rafael},
title = {{Gap MCSP Is Not (Levin) NP-Complete in Obfustopia}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {36:1--36:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.36},
URN = {urn:nbn:de:0030-drops-204322},
doi = {10.4230/LIPIcs.CCC.2024.36},
annote = {Keywords: Kolmogorov complexity, MCSP, Levin Reduction}
}