11 Search Results for "Ren, Hanlin"


Document
Total NP Search Problems with Abundant Solutions

Authors: Jiawei Li

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)


Abstract
We define a new complexity class TFAP to capture TFNP problems that possess abundant solutions for each input. We identify several problems across diverse fields that belong to TFAP, including WeakPigeon (finding a collision in a mapping from [2n] pigeons to [n] holes), Yamakawa-Zhandry’s problem [Takashi Yamakawa and Mark Zhandry, 2022], and all problems in TFZPP. Conversely, we introduce the notion of "semi-gluability" to characterize TFNP problems that could have a unique or a very limited number of solutions for certain inputs. We prove that there is no black-box reduction from any "semi-gluable" problems to any TFAP problems. Furthermore, it can be extended to rule out randomized black-box reduction in most cases. We identify that the majority of common TFNP subclasses, including PPA, PPAD, PPADS, PPP, PLS, CLS, SOPL, and UEOPL, are "semi-gluable". This leads to a broad array of oracle separation results within TFNP regime. As a corollary, UEOPL^O ⊈ PWPP^O relative to an oracle O.

Cite as

Jiawei Li. Total NP Search Problems with Abundant Solutions. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 75:1-75:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{li:LIPIcs.ITCS.2024.75,
  author =	{Li, Jiawei},
  title =	{{Total NP Search Problems with Abundant Solutions}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{75:1--75:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.75},
  URN =		{urn:nbn:de:0030-drops-196031},
  doi =		{10.4230/LIPIcs.ITCS.2024.75},
  annote =	{Keywords: TFNP, Pigeonhole Principle}
}
Document
Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness

Authors: Iddo Tzameret and Lu-Ming Zhang

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)


Abstract
We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich’s original work [S. Rudich, 1997], and apply it to draw new consequences in average-case complexity and proof complexity. Specifically, we show the following: Demi-bit stretch: Super-bits and demi-bits are variants of cryptographic pseudorandom generators which are secure against nondeterministic statistical tests [S. Rudich, 1997]. They were introduced to rule out certain approaches to proving strong complexity lower bounds beyond the limitations set out by the Natural Proofs barrier of Razborov and Rudich [A. A. Razborov and S. Rudich, 1997]. Whether demi-bits are stretchable at all had been an open problem since their introduction. We answer this question affirmatively by showing that: every demi-bit b:{0,1}ⁿ → {0,1}^{n+1} can be stretched into sublinear many demi-bits b':{0,1}ⁿ → {0,1}^{n+n^{c}}, for every constant 0 < c < 1. Average-case hardness: Using work by Santhanam [Rahul Santhanam, 2020], we apply our results to obtain new average-case Kolmogorov complexity results: we show that K^{poly}[n-O(1)] is zero-error average-case hard against NP/poly machines iff K^{poly}[n-o(n)] is, where for a function s(n):ℕ → ℕ, K^{poly}[s(n)] denotes the languages of all strings x ∈ {0,1}ⁿ for which there are (fixed) polytime Turing machines of description-length at most s(n) that output x. Characterising super-bits by nondeterministic unpredictability: In the deterministic setting, Yao [Yao, 1982] proved that super-polynomial hardness of pseudorandom generators is equivalent to ("next-bit") unpredictability. Unpredictability roughly means that given any strict prefix of a random string, it is infeasible to predict the next bit. We initiate the study of unpredictability beyond the deterministic setting (in the cryptographic regime), and characterise the nondeterministic hardness of generators from an unpredictability perspective. Specifically, we propose four stronger notions of unpredictability: NP/poly-unpredictability, coNP/poly-unpredictability, ∩-unpredictability and ∪-unpredictability, and show that super-polynomial nondeterministic hardness of generators lies between ∩-unpredictability and ∪-unpredictability. Characterising super-bits by nondeterministic hard-core predicates: We introduce a nondeterministic variant of hard-core predicates, called super-core predicates. We show that the existence of a super-bit is equivalent to the existence of a super-core of some non-shrinking function. This serves as an analogue of the equivalence between the existence of a strong pseudorandom generator and the existence of a hard-core of some one-way function [Goldreich and Levin, 1989; Håstad et al., 1999], and provides a first alternative characterisation of super-bits. We also prove that a certain class of functions, which may have hard-cores, cannot possess any super-core.

Cite as

Iddo Tzameret and Lu-Ming Zhang. Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 95:1-95:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{tzameret_et_al:LIPIcs.ITCS.2024.95,
  author =	{Tzameret, Iddo and Zhang, Lu-Ming},
  title =	{{Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{95:1--95:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.95},
  URN =		{urn:nbn:de:0030-drops-196234},
  doi =		{10.4230/LIPIcs.ITCS.2024.95},
  annote =	{Keywords: Pseudorandomness, Cryptography, Natural Proofs, Nondeterminism, Lower bounds}
}
Document
Bounded Relativization

Authors: Shuichi Hirahara, Zhenjian Lu, and Hanlin Ren

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)


Abstract
Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization. For a complexity class ℭ, we say that a statement is ℭ-relativizing if the statement holds relative to every oracle 𝒪 ∈ ℭ. It is easy to see that every result that relativizes also ℭ-relativizes for every complexity class ℭ. On the other hand, we observe that many non-relativizing results, such as IP = PSPACE, are in fact PSPACE-relativizing. First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ε > 0, BPE^{MCSP}/2^{εn} ⊈ SIZE[2ⁿ/n]. We prove this by PSPACE-relativizing the recent pseudodeterministic pseudorandom generator by Lu, Oliveira, and Santhanam (STOC 2021). Next, we study the limitations of PSPACE-relativizing proof techniques, and show that a seemingly minor improvement over the known results using PSPACE-relativizing techniques would imply a breakthrough separation NP ≠ L. For example: - Impagliazzo and Wigderson (JCSS 2001) proved that if EXP ≠ BPP, then BPP admits infinitely-often subexponential-time heuristic derandomization. We show that their result is PSPACE-relativizing, and that improving it to worst-case derandomization using PSPACE-relativizing techniques implies NP ≠ L. - Oliveira and Santhanam (STOC 2017) recently proved that every dense subset in P admits an infinitely-often subexponential-time pseudodeterministic construction, which we observe is PSPACE-relativizing. Improving this to almost-everywhere (pseudodeterministic) or (infinitely-often) deterministic constructions by PSPACE-relativizing techniques implies NP ≠ L. - Santhanam (SICOMP 2009) proved that pr-MA does not have fixed polynomial-size circuits. This lower bound can be shown PSPACE-relativizing, and we show that improving it to an almost-everywhere lower bound using PSPACE-relativizing techniques implies NP ≠ L. In fact, we show that if we can use PSPACE-relativizing techniques to obtain the above-mentioned improvements, then PSPACE ≠ EXPH. We obtain our barrier results by constructing suitable oracles computable in EXPH relative to which these improvements are impossible.

Cite as

Shuichi Hirahara, Zhenjian Lu, and Hanlin Ren. Bounded Relativization. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 6:1-6:45, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{hirahara_et_al:LIPIcs.CCC.2023.6,
  author =	{Hirahara, Shuichi and Lu, Zhenjian and Ren, Hanlin},
  title =	{{Bounded Relativization}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{6:1--6:45},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.6},
  URN =		{urn:nbn:de:0030-drops-182764},
  doi =		{10.4230/LIPIcs.CCC.2023.6},
  annote =	{Keywords: relativization, circuit lower bound, derandomization, explicit construction, pseudodeterministic algorithms, interactive proofs}
}
Document
An Algorithmic Approach to Uniform Lower Bounds

Authors: Rahul Santhanam

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)


Abstract
We propose a new family of circuit-based sampling tasks, such that non-trivial algorithmic solutions to certain tasks from this family imply frontier uniform lower bounds such as "NP is not in uniform ACC⁰" and "NP does not have uniform polynomial-size depth-two threshold circuits". Indeed, the most general versions of our sampling tasks have implications for central open problems such as NP vs P and PSPACE vs P. We argue the soundness of our approach by showing that the non-trivial algorithmic solutions we require do follow from standard cryptographic assumptions. In addition, we give evidence that a version of our approach for uniform circuits is necessary in order to separate NP from P or PSPACE from P. We give an algorithmic characterization for the PSPACE vs P question: PSPACE ≠ P iff either E has sub-exponential time non-uniform algorithms infinitely often or there are non-trivial space-efficient solutions to our sampling tasks for uniform Boolean circuits. We show how to use our framework to capture uniform versions of known non-uniform lower bounds, as well as classical uniform lower bounds such as the space hierarchy theorem and Allender’s uniform lower bound for the Permanent. We also apply our framework to prove new lower bounds: NP does not have polynomial-size uniform AC⁰ circuits with a bottom layer of MOD 6 gates, nor does it have polynomial-size uniform AC⁰ circuits with a bottom layer of threshold gates. Our proofs exploit recently defined probabilistic time-bounded variants of Kolmogorov complexity [Zhenjian Lu et al., 2022; Halley Goldberg et al., 2022; Halley Goldberg et al., 2022].

Cite as

Rahul Santhanam. An Algorithmic Approach to Uniform Lower Bounds. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 35:1-35:26, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{santhanam:LIPIcs.CCC.2023.35,
  author =	{Santhanam, Rahul},
  title =	{{An Algorithmic Approach to Uniform Lower Bounds}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{35:1--35:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.35},
  URN =		{urn:nbn:de:0030-drops-183053},
  doi =		{10.4230/LIPIcs.CCC.2023.35},
  annote =	{Keywords: Probabilistic Kolmogorov complexity, sampling algorithms, uniform lower bounds}
}
Document
New Lower Bounds and Derandomization for ACC, and a Derandomization-Centric View on the Algorithmic Method

Authors: Lijie Chen

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)


Abstract
In this paper, we obtain several new results on lower bounds and derandomization for ACC⁰ circuits (constant-depth circuits consisting of AND/OR/MOD_m gates for a fixed constant m, a frontier class in circuit complexity): 1) We prove that any polynomial-time Merlin-Arthur proof system with an ACC⁰ verifier (denoted by MA_{ACC⁰}) can be simulated by a nondeterministic proof system with quasi-polynomial running time and polynomial proof length, on infinitely many input lengths. This improves the previous simulation by [Chen, Lyu, and Williams, FOCS 2020], which requires both quasi-polynomial running time and proof length. 2) We show that MA_{ACC⁰} cannot be computed by fixed-polynomial-size ACC⁰ circuits, and our hard languages are hard on a sufficiently dense set of input lengths. 3) We show that NEXP (nondeterministic exponential-time) does not have ACC⁰ circuits of sub-half-exponential size, improving the previous sub-third-exponential size lower bound for NEXP against ACC⁰ by [Williams, J. ACM 2014]. Combining our first and second results gives a conceptually simpler and derandomization-centric proof of the recent breakthrough result NQP := NTIME[2^polylog(n)] ̸ ⊂ ACC⁰ by [Murray and Williams, SICOMP 2020]: Instead of going through an easy witness lemma as they did, we first prove an ACC⁰ lower bound for a subclass of MA, and then derandomize that subclass into NQP, while retaining its hardness against ACC⁰. Moreover, since our derandomization of MA_{ACC⁰} achieves a polynomial proof length, we indeed prove that nondeterministic quasi-polynomial-time with n^ω(1) nondeterminism bits (denoted as NTIMEGUESS[2^polylog(n), n^ω(1)]) has no poly(n)-size ACC⁰ circuits, giving a new proof of a result by Vyas. Combining with a win-win argument based on randomized encodings from [Chen and Ren, STOC 2020], we also prove that NTIMEGUESS[2^polylog(n), n^ω(1)] cannot be 1/2+1/poly(n)-approximated by poly(n)-size ACC⁰ circuits, improving the recent strongly average-case lower bounds for NQP against ACC⁰ by [Chen and Ren, STOC 2020]. One interesting technical ingredient behind our second result is the construction of a PSPACE-complete language that is paddable, downward self-reducible, same-length checkable, and weakly error correctable. Moreover, all its reducibility properties have corresponding AC⁰[2] non-adaptive oracle circuits. Our construction builds and improves upon similar constructions from [Trevisan and Vadhan, Complexity 2007] and [Chen, FOCS 2019], which all require at least TC⁰ oracle circuits for implementing these properties.

Cite as

Lijie Chen. New Lower Bounds and Derandomization for ACC, and a Derandomization-Centric View on the Algorithmic Method. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 34:1-34:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{chen:LIPIcs.ITCS.2023.34,
  author =	{Chen, Lijie},
  title =	{{New Lower Bounds and Derandomization for ACC, and a Derandomization-Centric View on the Algorithmic Method}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{34:1--34:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.34},
  URN =		{urn:nbn:de:0030-drops-175373},
  doi =		{10.4230/LIPIcs.ITCS.2023.34},
  annote =	{Keywords: Circuit Lower Bounds, Derandomization, Algorithmic Method, ACC}
}
Document
A Relativization Perspective on Meta-Complexity

Authors: Hanlin Ren and Rahul Santhanam

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)


Abstract
Meta-complexity studies the complexity of computational problems about complexity theory, such as the Minimum Circuit Size Problem (MCSP) and its variants. We show that a relativization barrier applies to many important open questions in meta-complexity. We give relativized worlds where: 1) MCSP can be solved in deterministic polynomial time, but the search version of MCSP cannot be solved in deterministic polynomial time, even approximately. In contrast, Carmosino, Impagliazzo, Kabanets, Kolokolova [CCC'16] gave a randomized approximate search-to-decision reduction for MCSP with a relativizing proof. 2) The complexities of MCSP[2^{n/2}] and MCSP[2^{n/4}] are different, in both worst-case and average-case settings. Thus the complexity of MCSP is not "robust" to the choice of the size function. 3) Levin’s time-bounded Kolmogorov complexity Kt(x) can be approximated to a factor (2+ε) in polynomial time, for any ε > 0. 4) Natural proofs do not exist, and neither do auxiliary-input one-way functions. In contrast, Santhanam [ITCS'20] gave a relativizing proof that the non-existence of natural proofs implies the existence of one-way functions under a conjecture about optimal hitting sets. 5) DistNP does not reduce to GapMINKT by a family of "robust" reductions. This presents a technical barrier for solving a question of Hirahara [FOCS'20].

Cite as

Hanlin Ren and Rahul Santhanam. A Relativization Perspective on Meta-Complexity. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 54:1-54:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{ren_et_al:LIPIcs.STACS.2022.54,
  author =	{Ren, Hanlin and Santhanam, Rahul},
  title =	{{A Relativization Perspective on Meta-Complexity}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{54:1--54:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.54},
  URN =		{urn:nbn:de:0030-drops-158646},
  doi =		{10.4230/LIPIcs.STACS.2022.54},
  annote =	{Keywords: meta-complexity, relativization, minimum circuit size problem}
}
Document
One-Way Functions and a Conditional Variant of MKTP

Authors: Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, and Ilya Volkovich

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)


Abstract
One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural NP-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows. 1) First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs. 2) Then, we observe that McKTP is NP-complete under polynomial-time randomized reductions. 3) Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP. Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem. In fact, building on recently-announced results of Ren and Santhanam [Rahul Ilango et al., 2021], we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs.

Cite as

Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, and Ilya Volkovich. One-Way Functions and a Conditional Variant of MKTP. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 7:1-7:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{allender_et_al:LIPIcs.FSTTCS.2021.7,
  author =	{Allender, Eric and Cheraghchi, Mahdi and Myrisiotis, Dimitrios and Tirumala, Harsha and Volkovich, Ilya},
  title =	{{One-Way Functions and a Conditional Variant of MKTP}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{7:1--7:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.7},
  URN =		{urn:nbn:de:0030-drops-155181},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.7},
  annote =	{Keywords: Kolmogorov complexity, KT Complexity, Minimum KT-complexity Problem, MKTP, Conditional KT Complexity, Minimum Conditional KT-complexity Problem, McKTP, one-way functions, OWFs, average-case hardness, pseudorandom generators, PRGs, pseudorandom functions, PRFs, distinguishers, learning algorithms, NP-completeness, reductions}
}
Document
Hardness of KT Characterizes Parallel Cryptography

Authors: Hanlin Ren and Rahul Santhanam

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)


Abstract
A recent breakthrough of Liu and Pass (FOCS'20) shows that one-way functions exist if and only if the (polynomial-)time-bounded Kolmogorov complexity, K^t, is bounded-error hard on average to compute. In this paper, we strengthen this result and extend it to other complexity measures: - We show, perhaps surprisingly, that the KT complexity is bounded-error average-case hard if and only if there exist one-way functions in constant parallel time (i.e. NC⁰). This result crucially relies on the idea of randomized encodings. Previously, a seminal work of Applebaum, Ishai, and Kushilevitz (FOCS'04; SICOMP'06) used the same idea to show that NC⁰-computable one-way functions exist if and only if logspace-computable one-way functions exist. - Inspired by the above result, we present randomized average-case reductions among the NC¹-versions and logspace-versions of K^t complexity, and the KT complexity. Our reductions preserve both bounded-error average-case hardness and zero-error average-case hardness. To the best of our knowledge, this is the first reduction between the KT complexity and a variant of K^t complexity. - We prove tight connections between the hardness of K^t complexity and the hardness of (the hardest) one-way functions. In analogy with the Exponential-Time Hypothesis and its variants, we define and motivate the Perebor Hypotheses for complexity measures such as K^t and KT. We show that a Strong Perebor Hypothesis for K^t implies the existence of (weak) one-way functions of near-optimal hardness 2^{n-o(n)}. To the best of our knowledge, this is the first construction of one-way functions of near-optimal hardness based on a natural complexity assumption about a search problem. - We show that a Weak Perebor Hypothesis for MCSP implies the existence of one-way functions, and establish a partial converse. This is the first unconditional construction of one-way functions from the hardness of MCSP over a natural distribution. - Finally, we study the average-case hardness of MKtP. We show that it characterizes cryptographic pseudorandomness in one natural regime of parameters, and complexity-theoretic pseudorandomness in another natural regime.

Cite as

Hanlin Ren and Rahul Santhanam. Hardness of KT Characterizes Parallel Cryptography. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 35:1-35:58, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{ren_et_al:LIPIcs.CCC.2021.35,
  author =	{Ren, Hanlin and Santhanam, Rahul},
  title =	{{Hardness of KT Characterizes Parallel Cryptography}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{35:1--35:58},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.35},
  URN =		{urn:nbn:de:0030-drops-143091},
  doi =		{10.4230/LIPIcs.CCC.2021.35},
  annote =	{Keywords: one-way function, meta-complexity, KT complexity, parallel cryptography, randomized encodings}
}
Document
Track A: Algorithms, Complexity and Games
Constructing a Distance Sensitivity Oracle in O(n^2.5794 M) Time

Authors: Yong Gu and Hanlin Ren

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)


Abstract
We continue the study of distance sensitivity oracles (DSOs). Given a directed graph G with n vertices and edge weights in {1, 2, … , M}, we want to build a data structure such that given any source vertex u, any target vertex v, and any failure f (which is either a vertex or an edge), it outputs the length of the shortest path from u to v not going through f. Our main result is a DSO with preprocessing time O(n^2.5794 M) and constant query time. Previously, the best preprocessing time of DSOs for directed graphs is O(n^2.7233 M), and even in the easier case of undirected graphs, the best preprocessing time is O(n^2.6865 M) [Ren, ESA 2020]. One drawback of our DSOs, though, is that it only supports distance queries but not path queries. Our main technical ingredient is an algorithm that computes the inverse of a degree-d polynomial matrix (i.e. a matrix whose entries are degree-d univariate polynomials) modulo x^r. The algorithm is adapted from [Zhou, Labahn and Storjohann, Journal of Complexity, 2015], and we replace some of its intermediate steps with faster rectangular matrix multiplication algorithms. We also show how to compute unique shortest paths in a directed graph with edge weights in {1, 2, … , M}, in O(n^2.5286 M) time. This algorithm is crucial in the preprocessing algorithm of our DSO. Our solution improves the O(n^2.6865 M) time bound in [Ren, ESA 2020], and matches the current best time bound for computing all-pairs shortest paths.

Cite as

Yong Gu and Hanlin Ren. Constructing a Distance Sensitivity Oracle in O(n^2.5794 M) Time. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 76:1-76:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{gu_et_al:LIPIcs.ICALP.2021.76,
  author =	{Gu, Yong and Ren, Hanlin},
  title =	{{Constructing a Distance Sensitivity Oracle in O(n^2.5794 M) Time}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{76:1--76:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.76},
  URN =		{urn:nbn:de:0030-drops-141450},
  doi =		{10.4230/LIPIcs.ICALP.2021.76},
  annote =	{Keywords: graph theory, shortest paths, distance sensitivity oracles}
}
Document
Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time

Authors: Hanlin Ren

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)


Abstract
We consider the problem of building Distance Sensitivity Oracles (DSOs). Given a directed graph G = (V, E) with edge weights in {1, 2, … , M}, we need to preprocess it into a data structure, and answer the following queries: given vertices u,v,x ∈ V, output the length of the shortest path from u to v that does not go through x. Our main result is a simple DSO with Õ(n^2.7233 M²) preprocessing time and O(1) query time. Moreover, if the input graph is undirected, the preprocessing time can be improved to Õ(n^2.6865 M²). Our algorithms are randomized with correct probability ≥ 1-1/n^c, for a constant c that can be made arbitrarily large. Previously, there is a DSO with Õ(n^2.8729 M) preprocessing time and polylog(n) query time [Chechik and Cohen, STOC'20]. At the core of our DSO is the following observation from [Bernstein and Karger, STOC'09]: if there is a DSO with preprocessing time P and query time Q, then we can construct a DSO with preprocessing time P+Õ(Mn²)⋅ Q and query time O(1). (Here Õ(⋅) hides polylog(n) factors.)

Cite as

Hanlin Ren. Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 79:1-79:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{ren:LIPIcs.ESA.2020.79,
  author =	{Ren, Hanlin},
  title =	{{Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{79:1--79:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.79},
  URN =		{urn:nbn:de:0030-drops-129450},
  doi =		{10.4230/LIPIcs.ESA.2020.79},
  annote =	{Keywords: Graph theory, Failure-prone structures}
}
Document
Approximating All-Pair Bounded-Leg Shortest Path and APSP-AF in Truly-Subcubic Time

Authors: Ran Duan and Hanlin Ren

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)


Abstract
In the bounded-leg shortest path (BLSP) problem, we are given a weighted graph G with nonnegative edge lengths, and we want to answer queries of the form "what's the shortest path from u to v, where only edges of length <=L are considered?". A more general problem is the APSP-AF (all-pair shortest path for all flows) problem, in which each edge has two weights - a length d and a capacity f, and a query asks about the shortest path from u to v where only edges of capacity >= f are considered. In this article we give an O~(n^{(omega+3)/2}epsilon^{-3/2}log W) time algorithm to compute a data structure that answers APSP-AF queries in O(log(epsilon^{-1}log (nW))) time and achieves (1+epsilon)-approximation, where omega < 2.373 is the exponent of time complexity of matrix multiplication, W is the upper bound of integer edge lengths, and n is the number of vertices. This is the first truly-subcubic time algorithm for these problems on dense graphs. Our algorithm utilizes the O(n^{(omega+3)/2}) time max-min product algorithm [Duan and Pettie 2009]. Since the all-pair bottleneck path (APBP) problem, which is equivalent to max-min product, can be seen as all-pair reachability for all flow, our approach indeed shows that these problems are almost equivalent in the approximation sense.

Cite as

Ran Duan and Hanlin Ren. Approximating All-Pair Bounded-Leg Shortest Path and APSP-AF in Truly-Subcubic Time. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 42:1-42:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{duan_et_al:LIPIcs.ICALP.2018.42,
  author =	{Duan, Ran and Ren, Hanlin},
  title =	{{Approximating All-Pair Bounded-Leg Shortest Path and APSP-AF in Truly-Subcubic Time}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{42:1--42:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.42},
  URN =		{urn:nbn:de:0030-drops-90467},
  doi =		{10.4230/LIPIcs.ICALP.2018.42},
  annote =	{Keywords: Graph Theory, Approximation Algorithms, Combinatorial Optimization}
}
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