15 Search Results for "Ilango, Rahul"

Document
DOI: 10.4230/LIPIcs.CCC.2024.29
Exact Search-To-Decision Reductions for Time-Bounded Kolmogorov Complexity

Authors: Shuichi Hirahara, Valentine Kabanets, Zhenjian Lu, and Igor C. Oliveira

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

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
DOI: 10.4230/LIPIcs.CCC.2024.34
Search-To-Decision Reductions for Kolmogorov Complexity

Authors: Noam Mazor and Rafael Pass

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

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
DOI: 10.4230/LIPIcs.CCC.2024.36
Gap MCSP Is Not (Levin) NP-Complete in Obfustopia

Authors: Noam Mazor and Rafael Pass

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

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}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.16
NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials

Authors: Omkar Baraskar, Agrim Dewan, Chandan Saha, and Pulkit Sinha

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
An s-sparse polynomial has at most s monomials with nonzero coefficients. The Equivalence Testing problem for sparse polynomials (ETsparse) asks to decide if a given polynomial f is equivalent to (i.e., in the orbit of) some s-sparse polynomial. In other words, given f ∈ 𝔽[𝐱] and s ∈ ℕ, ETsparse asks to check if there exist A ∈ GL(|𝐱|, 𝔽) and 𝐛 ∈ 𝔽^|𝐱| such that f(A𝐱 + 𝐛) is s-sparse. We show that ETsparse is NP-hard over any field 𝔽, if f is given in the sparse representation, i.e., as a list of nonzero coefficients and exponent vectors. This answers a question posed by Gupta, Saha and Thankey (SODA 2023) and also, more explicitly, by Baraskar, Dewan and Saha (STACS 2024). The result implies that the Minimum Circuit Size Problem (MCSP) is NP-hard for a dense subclass of depth-3 arithmetic circuits if the input is given in sparse representation. We also show that approximating the smallest s₀ such that a given s-sparse polynomial f is in the orbit of some s₀-sparse polynomial to within a factor of s^{1/3 - ε} is NP-hard for any ε > 0; observe that s-factor approximation is trivial as the input is s-sparse. Finally, we show that for any constant σ ≥ 6, checking if a polynomial (given in sparse representation) is in the orbit of some support-σ polynomial is NP-hard. Support of a polynomial f is the maximum number of variables present in any monomial of f. These results are obtained via direct reductions from the 3-SAT problem.
Cite as

Omkar Baraskar, Agrim Dewan, Chandan Saha, and Pulkit Sinha. NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{baraskar_et_al:LIPIcs.ICALP.2024.16,
  author =	{Baraskar, Omkar and Dewan, Agrim and Saha, Chandan and Sinha, Pulkit},
  title =	{{NP-Hardness of Testing Equivalence to Sparse Polynomials and to Constant-Support Polynomials}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{16:1--16:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.16},
  URN =		{urn:nbn:de:0030-drops-201598},
  doi =		{10.4230/LIPIcs.ICALP.2024.16},
  annote =	{Keywords: Equivalence testing, MCSP, sparse polynomials, 3SAT}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.110
Impagliazzo’s Worlds Through the Lens of Conditional Kolmogorov Complexity

Authors: Zhenjian Lu and Rahul Santhanam

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We develop new characterizations of Impagliazzo’s worlds Algorithmica, Heuristica and Pessiland by the intractability of conditional Kolmogorov complexity 𝖪 and conditional probabilistic time-bounded Kolmogorov complexity pK^t. In our first set of results, we show that NP ⊆ BPP iff pK^t(x ∣ y) can be computed efficiently in the worst case when t is sublinear in |x| + |y|; DistNP ⊆ HeurBPP iff pK^t(x ∣ y) can be computed efficiently over all polynomial-time samplable distributions when t is sublinear in |x| + |y|; and infinitely-often one-way functions fail to exist iff pK^t(x ∣ y) can be computed efficiently over all polynomial-time samplable distributions for t a sufficiently large polynomial in |x| + |y|. These results characterize Impagliazzo’s worlds Algorithmica, Heuristica and Pessiland purely in terms of the tractability of conditional pK^t. Notably, the results imply that Pessiland fails to exist iff the average-case intractability of conditional pK^t is insensitive to the difference between sublinear and polynomially bounded t. As a corollary, while we prove conditional pK^t to be NP-hard for sublinear t, showing NP-hardness for large enough polynomially bounded t would eliminate Pessiland as a possible world of average-case complexity. In our second set of results, we characterize Impagliazzo’s worlds Algorithmica, Heuristica and Pessiland by the distributional tractability of a natural problem, i.e., approximating the conditional Kolmogorov complexity, that is provably intractable in the worst case. We show that NP ⊆ BPP iff conditional Kolmogorov complexity can be approximated in the semi-worst case; and DistNP ⊆ HeurBPP iff conditional Kolmogorov complexity can be approximated on average over all independent polynomial-time samplable distributions. It follows from a result by Ilango, Ren, and Santhanam (STOC 2022) that infinitely-often one-way functions fail to exist iff conditional Kolmogorov complexity can be approximated on average over all polynomial-time samplable distributions. Together, these results yield the claimed characterizations. Our techniques, combined with previous work, also yield a characterization of auxiliary-input one-way functions and equivalences between different average-case tractability assumptions for conditional Kolmogorov complexity and its variants. Our results suggest that novel average-case tractability assumptions such as tractability in the semi-worst case and over independent polynomial-time samplable distributions might be worthy of further study.
Cite as

Zhenjian Lu and Rahul Santhanam. Impagliazzo’s Worlds Through the Lens of Conditional Kolmogorov Complexity. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 110:1-110:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lu_et_al:LIPIcs.ICALP.2024.110,
  author =	{Lu, Zhenjian and Santhanam, Rahul},
  title =	{{Impagliazzo’s Worlds Through the Lens of Conditional Kolmogorov Complexity}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{110:1--110:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.110},
  URN =		{urn:nbn:de:0030-drops-202538},
  doi =		{10.4230/LIPIcs.ICALP.2024.110},
  annote =	{Keywords: meta-complexity, Kolmogorov complexity, one-way functions, average-case complexity}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.29
On Randomized Reductions to the Random Strings

Authors: Michael Saks and Rahul Santhanam

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract
We study the power of randomized polynomial-time non-adaptive reductions to the problem of approximating Kolmogorov complexity and its polynomial-time bounded variants. As our first main result, we give a sharp dichotomy for randomized non-adaptive reducibility to approximating Kolmogorov complexity. We show that any computable language L that has a randomized polynomial-time non-adaptive reduction (satisfying a natural honesty condition) to ω(log(n))-approximating the Kolmogorov complexity is in AM ∩ coAM. On the other hand, using results of Hirahara [Shuichi Hirahara, 2020], it follows that every language in NEXP has a randomized polynomial-time non-adaptive reduction (satisfying the same honesty condition as before) to O(log(n))-approximating the Kolmogorov complexity. As our second main result, we give the first negative evidence against the NP-hardness of polynomial-time bounded Kolmogorov complexity with respect to randomized reductions. We show that for every polynomial t', there is a polynomial t such that if there is a randomized time t' non-adaptive reduction (satisfying a natural honesty condition) from SAT to ω(log(n))-approximating K^t complexity, then either NE = coNE or 𝖤 has sub-exponential size non-deterministic circuits infinitely often.
Cite as

Michael Saks and Rahul Santhanam. On Randomized Reductions to the Random Strings. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 29:1-29:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{saks_et_al:LIPIcs.CCC.2022.29,
  author =	{Saks, Michael and Santhanam, Rahul},
  title =	{{On Randomized Reductions to the Random Strings}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{29:1--29:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.29},
  URN =		{urn:nbn:de:0030-drops-165912},
  doi =		{10.4230/LIPIcs.CCC.2022.29},
  annote =	{Keywords: Kolmogorov complexity, randomized reductions}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.36
On One-Way Functions from NP-Complete Problems

Authors: Yanyi Liu and Rafael Pass

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract
We present the first natural NP-complete problem whose average-case hardness w.r.t. the uniform distribution over instances is equivalent to the existence of one-way functions (OWFs). The problem, which originated in the 1960s, is the Conditional Time-Bounded Kolmogorov Complexity Problem: let K^t(x∣z) be the length of the shortest "program" that, given the "auxiliary input" z, outputs the string x within time t(|x|), and let McK^tP[ζ] be the set of strings (x,z,k) where |z| = ζ(|x|), |k| = log |x| and K^t(x∣z) < k, where, for our purposes, a "program" is defined as a RAM machine. Our main result shows that for every polynomial t(n) ≥ n², there exists some polynomial ζ such that McK^tP[ζ] is NP-complete. We additionally extend the result of Liu-Pass (FOCS'20) to show that for every polynomial t(n) ≥ 1.1n, and every polynomial ζ(⋅), mild average-case hardness of McK^tP[ζ] is equivalent to the existence of OWFs. Taken together, these results provide the following crisp characterization of what is required to base OWFs on NP ⊈ BPP: There exists concrete polynomials t,ζ such that "Basing OWFs on NP ⊈ BPP" is equivalent to providing a "worst-case to (mild) average-case reduction for McK^tP[ζ]". In other words, the "holy-grail" of Cryptography (i.e., basing OWFs on NP ⊈ BPP) is equivalent to a basic question in algorithmic information theory. As an independent contribution, we show that our NP-completeness result can be used to shed new light on the feasibility of the polynomial-time bounded symmetry of information assertion (Kolmogorov'68).
Cite as

Yanyi Liu and Rafael Pass. On One-Way Functions from NP-Complete Problems. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 36:1-36:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{liu_et_al:LIPIcs.CCC.2022.36,
  author =	{Liu, Yanyi and Pass, Rafael},
  title =	{{On One-Way Functions from NP-Complete Problems}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{36:1--36:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.36},
  URN =		{urn:nbn:de:0030-drops-165981},
  doi =		{10.4230/LIPIcs.CCC.2022.36},
  annote =	{Keywords: One-way Functions, NP-Completeness, Kolmogorov Complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.45
Average-Case Hardness of NP and PH from Worst-Case Fine-Grained Assumptions

Authors: Lijie Chen, Shuichi Hirahara, and Neekon Vafa

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
What is a minimal worst-case complexity assumption that implies non-trivial average-case hardness of NP or PH? This question is well motivated by the theory of fine-grained average-case complexity and fine-grained cryptography. In this paper, we show that several standard worst-case complexity assumptions are sufficient to imply non-trivial average-case hardness of NP or PH: - NTIME[n] cannot be solved in quasi-linear time on average if UP ̸ ⊆ DTIME[2^{Õ(√n)}]. - Σ₂TIME[n] cannot be solved in quasi-linear time on average if Σ_kSAT cannot be solved in time 2^{Õ(√n)} for some constant k. Previously, it was not known if even average-case hardness of Σ₃SAT implies the average-case hardness of Σ₂TIME[n]. - Under the Exponential-Time Hypothesis (ETH), there is no average-case n^{1+ε}-time algorithm for NTIME[n] whose running time can be estimated in time n^{1+ε} for some constant ε > 0. Our results are given by generalizing the non-black-box worst-case-to-average-case connections presented by Hirahara (STOC 2021) to the settings of fine-grained complexity. To do so, we construct quite efficient complexity-theoretic pseudorandom generators under the assumption that the nondeterministic linear time is easy on average, which may be of independent interest.
Cite as

Lijie Chen, Shuichi Hirahara, and Neekon Vafa. Average-Case Hardness of NP and PH from Worst-Case Fine-Grained Assumptions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2022.45,
  author =	{Chen, Lijie and Hirahara, Shuichi and Vafa, Neekon},
  title =	{{Average-Case Hardness of NP and PH from Worst-Case Fine-Grained Assumptions}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{45:1--45:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.45},
  URN =		{urn:nbn:de:0030-drops-156411},
  doi =		{10.4230/LIPIcs.ITCS.2022.45},
  annote =	{Keywords: Average-case complexity, worst-case to average-case reduction}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2021.7
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
DOI: 10.4230/LIPIcs.CCC.2021.31
Hardness of Constant-Round Communication Complexity

Authors: Shuichi Hirahara, Rahul Ilango, and Bruno Loff

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

Abstract
How difficult is it to compute the communication complexity of a two-argument total Boolean function f:[N]×[N] → {0,1}, when it is given as an N×N binary matrix? In 2009, Kushilevitz and Weinreb showed that this problem is cryptographically hard, but it is still open whether it is NP-hard. In this work, we show that it is NP-hard to approximate the size (number of leaves) of the smallest constant-round protocol for a two-argument total Boolean function f:[N]×[N] → {0,1}, when it is given as an N×N binary matrix. Along the way to proving this, we show a new deterministic variant of the round elimination lemma, which may be of independent interest.
Cite as

Shuichi Hirahara, Rahul Ilango, and Bruno Loff. Hardness of Constant-Round Communication Complexity. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 31:1-31:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{hirahara_et_al:LIPIcs.CCC.2021.31,
  author =	{Hirahara, Shuichi and Ilango, Rahul and Loff, Bruno},
  title =	{{Hardness of Constant-Round Communication Complexity}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{31:1--31:30},
  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.2021.31},
  URN =		{urn:nbn:de:0030-drops-143055},
  doi =		{10.4230/LIPIcs.CCC.2021.31},
  annote =	{Keywords: NP-completeness, Communication Complexity, Round Elimination Lemma, Meta-Complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.94
An Efficient Coding Theorem via Probabilistic Representations and Its Applications

Authors: Zhenjian Lu and Igor C. Oliveira

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

Abstract
A probabilistic representation of a string x ∈ {0,1}ⁿ is given by the code of a randomized algorithm that outputs x with high probability [Igor C. Oliveira, 2019]. We employ probabilistic representations to establish the first unconditional Coding Theorem in time-bounded Kolmogorov complexity. More precisely, we show that if a distribution ensemble 𝒟_m can be uniformly sampled in time T(m) and generates a string x ∈ {0,1}^* with probability at least δ, then x admits a time-bounded probabilistic representation of complexity O(log(1/δ) + log (T) + log(m)). Under mild assumptions, a representation of this form can be computed from x and the code of the sampler in time polynomial in n = |x|. We derive consequences of this result relevant to the study of data compression, pseudodeterministic algorithms, time hierarchies for sampling distributions, and complexity lower bounds. In particular, we describe an instance-based search-to-decision reduction for Levin’s Kt complexity [Leonid A. Levin, 1984] and its probabilistic analogue rKt [Igor C. Oliveira, 2019]. As a consequence, if a string x admits a succinct time-bounded representation, then a near-optimal representation can be generated from x with high probability in polynomial time. This partially addresses in a time-bounded setting a question from [Leonid A. Levin, 1984] on the efficiency of computing an optimal encoding of a string.
Cite as

Zhenjian Lu and Igor C. Oliveira. An Efficient Coding Theorem via Probabilistic Representations and Its Applications. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 94:1-94:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{lu_et_al:LIPIcs.ICALP.2021.94,
  author =	{Lu, Zhenjian and Oliveira, Igor C.},
  title =	{{An Efficient Coding Theorem via Probabilistic Representations and Its Applications}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{94:1--94:20},
  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.2021.94},
  URN =		{urn:nbn:de:0030-drops-141630},
  doi =		{10.4230/LIPIcs.ICALP.2021.94},
  annote =	{Keywords: computational complexity, randomized algorithms, Kolmogorov complexity}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.22
NP-Hardness of Circuit Minimization for Multi-Output Functions

Authors: Rahul Ilango, Bruno Loff, and Igor C. Oliveira

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n → {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless 𝖯 = 𝖭𝖯, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions.
Cite as

Rahul Ilango, Bruno Loff, and Igor C. Oliveira. NP-Hardness of Circuit Minimization for Multi-Output Functions. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 22:1-22:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ilango_et_al:LIPIcs.CCC.2020.22,
  author =	{Ilango, Rahul and Loff, Bruno and Oliveira, Igor C.},
  title =	{{NP-Hardness of Circuit Minimization for Multi-Output Functions}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{22:1--22:36},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.22},
  URN =		{urn:nbn:de:0030-drops-125744},
  doi =		{10.4230/LIPIcs.CCC.2020.22},
  annote =	{Keywords: MCSP, circuit minimization, communication complexity, Boolean circuit}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.31
Connecting Perebor Conjectures: Towards a Search to Decision Reduction for Minimizing Formulas

Authors: Rahul Ilango

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
A longstanding open question is whether there is an equivalence between the computational task of determining the minimum size of any circuit computing a given function and the task of producing a minimum-sized circuit for a given function. While it is widely conjectured that both tasks require "perebor," or brute-force search, researchers have not yet ruled out the possibility that the search problem requires exponential time but the decision problem has a linear time algorithm. In this paper, we make progress in connecting the search and decision complexity of minimizing formulas. Let MFSP denote the problem that takes as input the truth table of a Boolean function f and an integer size parameter s and decides whether there is a formula for f of size at most s. Let Search- denote the corresponding search problem where one has to output some optimal formula for computing f. Our main result is that given an oracle to MFSP, one can solve Search-MFSP in time polynomial in the length N of the truth table of f and the number t of "near-optimal" formulas for f, in particular O(N⁶t²)-time. While the quantity t is not well understood, we use this result (and some extensions) to prove that given an oracle to MFSP: - there is a deterministic 2^O(N/(log log N))-time oracle algorithm for solving Search-MFSP on all but a o(1)-fraction of instances, and - there is a randomized O(2^.67N)-time oracle algorithm for solving Search-MFSP on all instances. Intriguingly, the main idea behind our algorithms is in some sense a "reverse application" of the gate elimination technique.
Cite as

Rahul Ilango. Connecting Perebor Conjectures: Towards a Search to Decision Reduction for Minimizing Formulas. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 31:1-31:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ilango:LIPIcs.CCC.2020.31,
  author =	{Ilango, Rahul},
  title =	{{Connecting Perebor Conjectures: Towards a Search to Decision Reduction for Minimizing Formulas}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{31:1--31:35},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.31},
  URN =		{urn:nbn:de:0030-drops-125834},
  doi =		{10.4230/LIPIcs.CCC.2020.31},
  annote =	{Keywords: minimum circuit size problem, minimum formula size problem, gate elimination, search to decision reduction, self-reducibility}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.34
Approaching MCSP from Above and Below: Hardness for a Conditional Variant and AC^0[p]

Authors: Rahul Ilango

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
The Minimum Circuit Size Problem (MCSP) asks whether a given Boolean function has a circuit of at most a given size. MCSP has been studied for over a half-century and has deep connections throughout theoretical computer science including to cryptography, computational learning theory, and proof complexity. For example, we know (informally) that if MCSP is easy to compute, then most cryptography can be broken. Despite this cryptographic hardness connection and extensive research, we still know relatively little about the hardness of MCSP unconditionally. Indeed, until very recently it was unknown whether MCSP can be computed in AC^0[2] (Golovnev et al., ICALP 2019). Our main contribution in this paper is to formulate a new "oracle" variant of circuit complexity and prove that this problem is NP-complete under randomized reductions. In more detail, we define the Minimum Oracle Circuit Size Problem (MOCSP) that takes as input the truth table of a Boolean function f, a size threshold s, and the truth table of an oracle Boolean function O, and determines whether there is a circuit with O-oracle gates and at most s wires that computes f. We prove that MOCSP is NP-complete under randomized polynomial-time reductions. We also extend the recent AC^0[p] lower bound against MCSP by Golovnev et al. to a lower bound against the circuit minimization problem for depth-d formulas, (AC^0_d)-MCSP. We view this result as primarily a technical contribution. In particular, our proof takes a radically different approach from prior MCSP-related hardness results.
Cite as

Rahul Ilango. Approaching MCSP from Above and Below: Hardness for a Conditional Variant and AC^0[p]. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 34:1-34:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ilango:LIPIcs.ITCS.2020.34,
  author =	{Ilango, Rahul},
  title =	{{Approaching MCSP from Above and Below: Hardness for a Conditional Variant and AC^0\lbrackp\rbrack}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{34:1--34:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.34},
  URN =		{urn:nbn:de:0030-drops-117191},
  doi =		{10.4230/LIPIcs.ITCS.2020.34},
  annote =	{Keywords: Minimum Circuit Size Problem, reductions, NP-completeness, circuit lower bounds}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.66
AC^0[p] Lower Bounds Against MCSP via the Coin Problem

Authors: Alexander Golovnev, Rahul Ilango, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, and Avishay Tal

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
Minimum Circuit Size Problem (MCSP) asks to decide if a given truth table of an n-variate boolean function has circuit complexity less than a given parameter s. We prove that MCSP is hard for constant-depth circuits with mod p gates, for any prime p >= 2 (the circuit class AC^0[p]). Namely, we show that MCSP requires d-depth AC^0[p] circuits of size at least exp(N^{0.49/d}), where N=2^n is the size of an input truth table of an n-variate boolean function. Our circuit lower bound proof shows that MCSP can solve the coin problem: distinguish uniformly random N-bit strings from those generated using independent samples from a biased random coin which is 1 with probability 1/2+N^{-0.49}, and 0 otherwise. Solving the coin problem with such parameters is known to require exponentially large AC^0[p] circuits. Moreover, this also implies that MAJORITY is computable by a non-uniform AC^0 circuit of polynomial size that also has MCSP-oracle gates. The latter has a few other consequences for the complexity of MCSP, e.g., we get that any boolean function in NC^1 (i.e., computable by a polynomial-size formula) can also be computed by a non-uniform polynomial-size AC^0 circuit with MCSP-oracle gates.
Cite as

Alexander Golovnev, Rahul Ilango, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, and Avishay Tal. AC^0[p] Lower Bounds Against MCSP via the Coin Problem. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{golovnev_et_al:LIPIcs.ICALP.2019.66,
  author =	{Golovnev, Alexander and Ilango, Rahul and Impagliazzo, Russell and Kabanets, Valentine and Kolokolova, Antonina and Tal, Avishay},
  title =	{{AC^0\lbrackp\rbrack Lower Bounds Against MCSP via the Coin Problem}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{66:1--66:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.66},
  URN =		{urn:nbn:de:0030-drops-106422},
  doi =		{10.4230/LIPIcs.ICALP.2019.66},
  annote =	{Keywords: Minimum Circuit Size Problem (MCSP), circuit lower bounds, AC0\lbrackp\rbrack, coin problem, hybrid argument, MKTP, biased random boolean functions}
}
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