12 Search Results for "Gajulapalli, Karthik"

Document
DOI: 10.4230/LIPIcs.STACS.2026.28
Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank

Authors: Nikolai Chukhin, Alexander S. Kulikov, Ivan Mihajlin, and Arina Smirnova

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
Proving complexity lower bounds remains a challenging task: currently, we only know how to prove conditional uniform (algorithm) lower bounds and nonuniform (circuit) lower bounds in restricted circuit models. About a decade ago, Williams (STOC 2010) showed how to derive nonuniform lower bounds from uniform upper bounds: roughly, by designing a fast algorithm for checking satisfiability of circuits, one gets a lower bound for this circuit class. Since then, a number of results of this kind have been proved. For example, Jahanjou et al. (ICALP 2015) and Carmosino et al. (ITCS 2016) proved that if NSETH fails, then E^{NP} has series-parallel circuit size ω(n). One can also derive nonuniform lower bounds from nondeterministic uniform lower bounds. Perhaps the most well-known example is the Karp-Lipton theorem (STOC 1980): if Σ₂ ≠ Π₂, then NP ⊄ P/poly. Some recent examples include the following. Nederlof (STOC 2020) proved a lower bound on the matrix multiplication tensor rank under an assumption that TSP cannot be solved faster than in 2ⁿ time. Belova et al. (SODA 2024) proved that there exists an explicit polynomial family of arithmetic circuit size Ω(n^{δ}), for any δ > 0, assuming that MAX-3-SAT cannot be solved faster than in 2ⁿ nondeterministic time. Williams (FOCS 2024) proved an exponential lower bound for ETHR ∘ ETHR circuits under the Orthogonal Vectors conjecture. Whereas all the lower bounds above are proved under strong assumptions that might eventually be refuted, the revealed connections are of great interest and may still give further insights: one may be able to weaken the used assumptions or to construct generators from other fine-grained reductions. In this paper, we continue developing this line of research and show how uniform nondeterministic lower bounds can be used to construct generators of various types of combinatorial objects that are notoriously hard to analyze: Boolean functions of high circuit size, matrices of high rigidity, and tensors of high rank. Specifically, we prove the following. - If, for some ε and k, k-SAT cannot be solved in input-oblivious co-nondeterministic time O(2^{(1/2+ε)n}), then there exists a monotone Boolean function family in coNP of monotone circuit size 2^{Ω(n / log n)}. Combining this with the result above, we get win-win circuit lower bounds: either E^{NP{}} requires series-parallel circuits of size ω(n) or coNP requires monotone circuits of size 2^{Ω(n / log n)}. - If, for all ε > 0, MAX-3-SAT cannot be solved in co-nondeterministic time O(2^{(1 - ε)n}), then there exist small families of matrices with rigidity exceeding the best known constructions as well as small families of three-dimensional tensors of rank n^{1+Δ}, for some Δ > 0.
Cite as

Nikolai Chukhin, Alexander S. Kulikov, Ivan Mihajlin, and Arina Smirnova. Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chukhin_et_al:LIPIcs.STACS.2026.28,
  author =	{Chukhin, Nikolai and Kulikov, Alexander S. and Mihajlin, Ivan and Smirnova, Arina},
  title =	{{Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{28:1--28:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.28},
  URN =		{urn:nbn:de:0030-drops-255177},
  doi =		{10.4230/LIPIcs.STACS.2026.28},
  annote =	{Keywords: computational complexity, circuit complexity, lower bounds, conditional lower bounds, monotone circuits, matrix rigidity, tensor rank, arithmetic circuits, fine-grained complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.60
Total Search Problems in ZPP

Authors: Noah Fleming, Stefan Grosser, Siddhartha Jain, Jiawei Li, Hanlin Ren, Morgan Shirley, and Weiqiang Yuan

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
We initiate a systematic study of TFZPP, the class of total NP search problems solvable by polynomial time randomized algorithms. TFZPP contains a variety of important search problems such as Bertrand-Chebyshev (finding a prime between N and 2N), refuter problems for many circuit lower bounds, and Lossy-Code. The Lossy-Code problem has found prominence due to its fundamental connections to derandomization, catalytic computing, and the metamathematics of complexity theory, among other areas. While TFZPP collapses to FP under standard derandomization assumptions in the white-box setting, we are able to separate TFZPP from the major TFNP subclasses in the black-box setting. In fact, we are able to separate it from every uniform TFNP class assuming that NP is not in quasi-polynomial time. To do so, we extend the connection between proof complexity and black-box TFNP to randomized proof systems and randomized reductions. Next, we turn to developing a taxonomy of TFZPP problems. We highlight a problem called Nephew, originating from an infinity axiom in set theory. We show that Nephew is in PWPP∩ TFZPP and conjecture that it is not reducible to Lossy-Code. Intriguingly, except for some artificial examples, most other black-box TFZPP problems that we are aware of reduce to Lossy-Code: - We define a problem called Empty-Child capturing finding a leaf in a rooted (binary) tree, and show that this problem is equivalent to Lossy-Code. We also show that a variant of Empty-Child with "heights" is complete for the intersection of SOPL and Lossy-Code. - We strengthen Lossy-Code with several combinatorial inequalities such as the AM-GM inequality. Somewhat surprisingly, we show the resulting new problems are still reducible to Lossy-Code. A technical highlight of this result is that they are proved by formalizations in bounded arithmetic, specifically in Jeřábek’s theory APC₁ (JSL 2007). - Finally, we show that the Dense-Linear-Ordering problem reduces to Lossy-Code.
Cite as

Noah Fleming, Stefan Grosser, Siddhartha Jain, Jiawei Li, Hanlin Ren, Morgan Shirley, and Weiqiang Yuan. Total Search Problems in ZPP. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 60:1-60:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fleming_et_al:LIPIcs.ITCS.2026.60,
  author =	{Fleming, Noah and Grosser, Stefan and Jain, Siddhartha and Li, Jiawei and Ren, Hanlin and Shirley, Morgan and Yuan, Weiqiang},
  title =	{{Total Search Problems in ZPP}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{60:1--60:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.60},
  URN =		{urn:nbn:de:0030-drops-253473},
  doi =		{10.4230/LIPIcs.ITCS.2026.60},
  annote =	{Keywords: TFNP, lossy code, randomized proof systems, query complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.111
Hardness of Range Avoidance and Proof Complexity Generators from Demi-Bits

Authors: Hanlin Ren, Yichuan Wang, and Yan Zhong

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Given a circuit G: {0, 1}ⁿ → {0, 1}^m with m > n, the range avoidance problem (Avoid) asks to output a string y ∈ {0, 1}^m that is not in the range of G. Besides its profound connection to circuit complexity and explicit construction problems, this problem is also related to the existence of proof complexity generators - circuits G: {0, 1}ⁿ → {0, 1}^m where m > n but for every y ∈ {0, 1}^m, it is infeasible to prove the statement "y ̸ ∈ Range(G)" in a given propositional proof system. This paper connects these two problems with the existence of demi-bits generators, a fundamental cryptographic primitive against nondeterministic adversaries introduced by Rudich (RANDOM '97). - We show that the existence of demi-bits generators implies Avoid is hard for nondeterministic algorithms. This resolves an open problem raised by Chen and Li (STOC '24). Furthermore, assuming the demi-hardness of certain LPN-style generators or Goldreich’s PRG, we prove the hardness of Avoid even when the instances are constant-degree polynomials over 𝔽₂. - We show that the dual weak pigeonhole principle is unprovable in Cook’s theory PV₁ under the existence of demi-bits generators secure against AM/_{O(1)}, thereby separating Jeřábek’s theory APC₁ from PV₁. Previously, Ilango, Li, and Williams (STOC '23) obtained the same separation under different (and arguably stronger) cryptographic assumptions. - We transform demi-bits generators to proof complexity generators that are pseudo-surjective in certain parameter regime. Pseudo-surjectivity is the strongest form of hardness considered in the literature for proof complexity generators. Our constructions are inspired by the recent breakthroughs on the hardness of Avoid by Ilango, Li, and Williams (STOC '23) and Chen and Li (STOC '24). We use randomness extractors to significantly simplify the construction and the proof.
Cite as

Hanlin Ren, Yichuan Wang, and Yan Zhong. Hardness of Range Avoidance and Proof Complexity Generators from Demi-Bits. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 111:1-111:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ren_et_al:LIPIcs.ITCS.2026.111,
  author =	{Ren, Hanlin and Wang, Yichuan and Zhong, Yan},
  title =	{{Hardness of Range Avoidance and Proof Complexity Generators from Demi-Bits}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{111:1--111:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.111},
  URN =		{urn:nbn:de:0030-drops-253982},
  doi =		{10.4230/LIPIcs.ITCS.2026.111},
  annote =	{Keywords: Range Avoidance, Proof Complexity Generators}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.79
Range Avoidance and Remote Point: New Algorithms and Hardness

Authors: Shengtang Huang, Xin Li, and Yan Zhong

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The Range Avoidance (Avoid) problem C-Avoid[n,m(n)] asks that, given a circuit in a class C with input length n and output length m(n) > n, find a string not in the range of the circuit. This problem has been a central piece in several recent frameworks for proving circuit lower bounds and constructing explicit combinatorial objects. Previous work by Korten (FOCS' 21) and by Ren, Santhanam, and Wang (FOCS' 22) showed that algorithms for Avoid are closely related to circuit lower bounds. In particular, Korten’s work reinterpreted an earlier result from bounded arithmetic, originally proved by Jeřábek (Ann. Pure Appl. Log. 2004), as an equivalence in computational complexity between the existence of FP^NP algorithms for the general Avoid problem and 2^{Ω(n)} lower bounds against general Boolean circuits for the class 𝐄^NP. In this work, we significantly complement these works by generalizing the equivalence result to restricted circuit classes and obtain the following: - For any constant depth unbounded fan-in circuit class C ⊇ AC⁰, there is an FP^NP algorithm for C-Avoid[n,n^{1+ε}] (for any constant ε > 0) if and only if 𝐄^NP cannot be computed by C circuits of size 2^{o(n)}. This addresses an open problem by Korten (Bulletin of EATCS' 25). - If 𝐄^NP cannot be computed by o(2ⁿ/n) size formulas, then there is an FP^NP algorithm for NC⁰-Avoid[n,2n]. Note that by an extension of Ren, Santhanam, and Wang (FOCS' 22), an FP^NP algorithm for NC⁰₄-Avoid[n,n+n^δ] for any constant δ ∈ (0,1) implies 𝐄^NP cannot be computed by o(2ⁿ/n) size formulas. These results yield the first characterizations of FP^NP C-Avoid algorithms for low-complexity circuit classes such as AC⁰. We also consider the average-case analog of Avoid, the Remote Point (Remote-Point) problem, and establish: - For some suitable function c(n) and constant γ > 0, there is an FP^NP algorithm for Remote-Point[n,n^{6+γ},c(O_{γ}(log n))] if and only if 𝐄^NP cannot be (1/2-c(n))-approximated by circuits of size 2^{o(n)}. Finally, we also present two improved algorithms for NC⁰-Avoid: - A family of 2^{n^{1 - ε/(k-1) +o(1)}} time algorithms for NC⁰_k-Avoid[n,n^{1+ε}] for any ε > 0, exhibiting the first subexponential-time algorithm for any super-linear stretch. - Faster local algorithms for NC⁰_k-Avoid[n,n+1] running in time O(n2^{(k-2)/(k-1) n}), improving the naive 2ⁿ⋅ poly(n) bound.
Cite as

Shengtang Huang, Xin Li, and Yan Zhong. Range Avoidance and Remote Point: New Algorithms and Hardness. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 79:1-79:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{huang_et_al:LIPIcs.ITCS.2026.79,
  author =	{Huang, Shengtang and Li, Xin and Zhong, Yan},
  title =	{{Range Avoidance and Remote Point: New Algorithms and Hardness}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{79:1--79:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  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.ITCS.2026.79},
  URN =		{urn:nbn:de:0030-drops-253662},
  doi =		{10.4230/LIPIcs.ITCS.2026.79},
  annote =	{Keywords: Circuit Lower Bounds, Range Avoidance Problem, Remote Point Problem}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.62
Avoiding Range via Turan-Type Bounds

Authors: Neha Kuntewar and Jayalal Sarma

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
Given a circuit C : {0,1}^n → {0,1}^m from a circuit class 𝒞, with m > n, finding a y ∈ {0,1}^m such that ∀ x ∈ {0,1}ⁿ, C(x) ≠ y, is the range avoidance problem (denoted by C-Avoid). Deterministic polynomial time algorithms (even with access to NP oracles) solving this problem are known to imply explicit constructions of various pseudorandom objects like hard Boolean functions, linear codes, PRGs etc. Deterministic polynomial time algorithms are known for NC⁰₂-Avoid when m > n, and for NC⁰₃-Avoid when m ≥ n²/log n, where NC⁰_k is the class of circuits with bounded fan-in which have constant depth and the output depends on at most k of the input bits. On the other hand, it is also known that NC⁰₃-Avoid when m = n+O(n^{2/3}) is at least as hard as explicit construction of rigid matrices. In fact, algorithms for solving range avoidance for even NC⁰₄ circuits imply new circuit lower bounds. In this paper, we propose a new approach to solving the range avoidance problem via hypergraphs. We formulate the problem in terms of Turan-type problems in hypergraphs of the following kind: for a fixed k-uniform hypergraph H, what is the maximum number of edges that can exist in H_C, which does not have a sub-hypergraph isomorphic to H? We show the following: - We first demonstrate the applicability of this approach by showing alternate proofs of some of the known results for the range avoidance problem using this framework. - We then use our approach to show (using several different hypergraph structures for which Turan-type bounds are known in the literature) that there is a constant c such that Monotone-NC⁰₃-Avoid can be solved in deterministic polynomial time when m > cn². - To improve the stretch constraint to linear, more precisely, to m > n, we show a new Turan-type theorem for a hypergraph structure (which we call the loose X_{2ℓ}-cycles). More specifically, we prove that any connected 3-uniform linear hypergraph with m > n edges must contain a loose X_{2ℓ} cycle. This may be of independent interest. - Using this, we show that Monotone-NC⁰₃-Avoid can be solved in deterministic polynomial time when m > n, thus improving the known bounds of NC⁰₃-Avoid for the case of monotone circuits. In contrast, we note that efficient algorithms for solving Monotone-NC⁰₆-Avoid, already imply explicit constructions for rigid matrices. - We also generalise our argument to solve the special case of range avoidance for NC⁰_k where each output function computed by the circuit is the majority function on its inputs, where m > n².
Cite as

Neha Kuntewar and Jayalal Sarma. Avoiding Range via Turan-Type Bounds. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 62:1-62:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kuntewar_et_al:LIPIcs.APPROX/RANDOM.2025.62,
  author =	{Kuntewar, Neha and Sarma, Jayalal},
  title =	{{Avoiding Range via Turan-Type Bounds}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{62:1--62:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.62},
  URN =		{urn:nbn:de:0030-drops-244281},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.62},
  annote =	{Keywords: circuit lower bounds, explicit constructions, range avoidance, linear hypergraphs, Tur\'{a}n number of hypergraphs}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.35
How to Construct Random Strings

Authors: Oliver Korten and Rahul Santhanam

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
We address the following fundamental question: is there an efficient deterministic algorithm that, given 1ⁿ, outputs a string of length n that has polynomial-time bounded Kolmogorov complexity Ω̃(n) or even n - o(n)? Under plausible complexity-theoretic assumptions, stating for example that there is an ε > 0 for which TIME[T(n)] ̸ ⊆ TIME^NP[T(n)^ε]/2^(εn) for appropriately chosen time-constructible T, we show that the answer to this question is positive (answering a question of [Hanlin Ren et al., 2022]), and that the Range Avoidance problem [Robert Kleinberg et al., 2021; Oliver Korten, 2021; Hanlin Ren et al., 2022] is efficiently solvable for uniform sequences of circuits with close to minimal stretch (answering a question of [Rahul Ilango et al., 2023]). We obtain our results by giving efficient constructions of pseudo-random generators with almost optimal seed length against algorithms with small advice, under assumptions of the form mentioned above. We also apply our results to give the first complexity-theoretic evidence for explicit constructions of objects such as rigid matrices (in the sense of Valiant) and Ramsey graphs with near-optimal parameters.
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Oliver Korten and Rahul Santhanam. How to Construct Random Strings. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 35:1-35:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{korten_et_al:LIPIcs.CCC.2025.35,
  author =	{Korten, Oliver and Santhanam, Rahul},
  title =	{{How to Construct Random Strings}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{35:1--35:32},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.35},
  URN =		{urn:nbn:de:0030-drops-237290},
  doi =		{10.4230/LIPIcs.CCC.2025.35},
  annote =	{Keywords: Explicit Constructions, Kolmogorov Complexity, Derandomization}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2024.23
Oblivious Complexity Classes Revisited: Lower Bounds and Hierarchies

Authors: Karthik Gajulapalli, Zeyong Li, and Ilya Volkovich

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract
In this work we study oblivious complexity classes. These classes capture the power of interactive proofs where the prover(s) are only given the input size rather than the actual input. In particular, we study the connections between the symmetric polynomial time - S₂P and its oblivious counterpart - O₂P. Among our results: - For each k ∈ ℕ, we construct an explicit language L_k ∈ O₂P that cannot be computed by circuits of size n^k. - We prove a hierarchy theorem for O₂TIME. In particular, for any time constructible function t:ℕ → ℕ and any ε > 0 we show that: O₂TIME[t(n)] ⊊ O₂TIME[t(n)^{1 + ε}]. - We prove new structural results connecting O₂P and S₂P. - We make partial progress towards the resolution of an open question posed by Goldreich and Meir (TOCT 2015) that relates the complexity of NP to its oblivious counterpart - ONP. - We identify a natural class of problems in O₂P from computational Ramsey theory, that are not expected to be in 𝖯 or even BPP. To the best of our knowledge, these results constitute the first explicit fixed-polynomial lower bound and hierarchy theorem for O₂P. The smallest uniform complexity class for which such lower bounds were previously known was S₂P due to Cai (JCSS 2007). In addition, this is the first uniform hierarchy theorem for a semantic class. All previous results required some non-uniformity. In order to obtain some of the results in the paper, we introduce the notion of uniformly-sparse extensions which could be of independent interest. Our techniques build upon the de-randomization framework of the powerful Range Avoidance problem that has yielded many new interesting explicit circuit lower bounds.
Cite as

Karthik Gajulapalli, Zeyong Li, and Ilya Volkovich. Oblivious Complexity Classes Revisited: Lower Bounds and Hierarchies. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gajulapalli_et_al:LIPIcs.FSTTCS.2024.23,
  author =	{Gajulapalli, Karthik and Li, Zeyong and Volkovich, Ilya},
  title =	{{Oblivious Complexity Classes Revisited: Lower Bounds and Hierarchies}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{23:1--23:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.23},
  URN =		{urn:nbn:de:0030-drops-222122},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.23},
  annote =	{Keywords: fixed circuit lower bounds, semantic time hierarchy, oblivious complexity, range avoidance}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.42
Matrix Multiplication Verification Using Coding Theory

Authors: Huck Bennett, Karthik Gajulapalli, Alexander Golovnev, and Evelyn Warton

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three n × n matrices A, B, and C as input, to decide whether AB = C. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in Õ(n²) time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in o(n^ω) time). To that end, we give two algorithms for MMV in the case where AB - C is sparse. Specifically, when AB - C has at most O(n^δ) non-zero entries for a constant 0 ≤ δ < 2, we give (1) a deterministic O(n^(ω-ε))-time algorithm for constant ε = ε(δ) > 0, and (2) a randomized Õ(n²)-time algorithm using δ/2 ⋅ log₂ n + O(1) random bits. The former algorithm is faster than the deterministic algorithm of Künnemann (ESA, 2018) when δ ≥ 1.056, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses log₂ n + O(1) random bits (in turn fewer than Freivalds’s algorithm). Our algorithms are simple and use techniques from coding theory. Let H be a parity-check matrix of a Maximum Distance Separable (MDS) code, and let G = (I | G') be a generator matrix of a (possibly different) MDS code in systematic form. Our deterministic algorithm uses fast rectangular matrix multiplication to check whether HAB = HC and H(AB)^T = H(C^T), and our randomized algorithm samples a uniformly random row g' from G' and checks whether g'AB = g'C and g'(AB)^T = g'C^T. We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require Ω(n^ω) time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic Õ(n²)-time reductions).
Cite as

Huck Bennett, Karthik Gajulapalli, Alexander Golovnev, and Evelyn Warton. Matrix Multiplication Verification Using Coding Theory. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bennett_et_al:LIPIcs.APPROX/RANDOM.2024.42,
  author =	{Bennett, Huck and Gajulapalli, Karthik and Golovnev, Alexander and Warton, Evelyn},
  title =	{{Matrix Multiplication Verification Using Coding Theory}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{42:1--42:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.42},
  URN =		{urn:nbn:de:0030-drops-210352},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.42},
  annote =	{Keywords: Matrix Multiplication Verification, Derandomization, Sparse Matrices, Error-Correcting Codes, Hardness Barriers, Reductions}
}
Document
DOI: 10.4230/LIPIcs.ITC.2024.5
On the Power of Adaptivity for Function Inversion

Authors: Karthik Gajulapalli, Alexander Golovnev, and Samuel King

Published in: LIPIcs, Volume 304, 5th Conference on Information-Theoretic Cryptography (ITC 2024)

Abstract
We study the problem of function inversion with preprocessing where, given a function f : [N] → [N] and a point y in its image, the goal is to find an x such that f(x) = y using at most T oracle queries to f and S bits of preprocessed advice that depend on f. The seminal work of Corrigan-Gibbs and Kogan [TCC 2019] initiated a line of research that shows many exciting connections between the non-adaptive setting of this problem and other areas of theoretical computer science. Specifically, they introduced a very weak class of algorithms (strongly non-adaptive) where the points queried by the oracle depend only on the inversion point y, and are independent of the answers to the previous queries and the S bits of advice. They showed that proving even mild lower bounds on strongly non-adaptive algorithms for function inversion would imply a breakthrough result in circuit complexity. We prove that every strongly non-adaptive algorithm for function inversion (and even for its special case of permutation inversion) must have ST = Ω(N log (N) log (T)). This gives the first improvement to the long-standing lower bound of ST = Ω(N log N) due to Yao [STOC 90]. As a corollary, we conclude the first separation between strongly non-adaptive and adaptive algorithms for permutation inversion, where the adaptive algorithm by Hellman [TOIT 80] achieves the trade-off ST = O(N log N). Additionally, we show equivalence between lower bounds for strongly non-adaptive data structures and the one-way communication complexity of certain partial functions. As an example, we recover our lower bound on function inversion in the communication complexity framework.
Cite as

Karthik Gajulapalli, Alexander Golovnev, and Samuel King. On the Power of Adaptivity for Function Inversion. In 5th Conference on Information-Theoretic Cryptography (ITC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 304, pp. 5:1-5:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gajulapalli_et_al:LIPIcs.ITC.2024.5,
  author =	{Gajulapalli, Karthik and Golovnev, Alexander and King, Samuel},
  title =	{{On the Power of Adaptivity for Function Inversion}},
  booktitle =	{5th Conference on Information-Theoretic Cryptography (ITC 2024)},
  pages =	{5:1--5:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-333-1},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{304},
  editor =	{Aggarwal, Divesh},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2024.5},
  URN =		{urn:nbn:de:0030-drops-205137},
  doi =		{10.4230/LIPIcs.ITC.2024.5},
  annote =	{Keywords: Function Inversion, Non-Adaptive lower bounds, Communication Complexity}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.30
Rapid Mixing for the Hardcore Glauber Dynamics and Other Markov Chains in Bounded-Treewidth Graphs

Authors: David Eppstein and Daniel Frishberg

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
We give a new rapid mixing result for a natural random walk on the independent sets of a graph G. We show that when G has bounded treewidth, this random walk - known as the Glauber dynamics for the hardcore model - mixes rapidly for all fixed values of the standard parameter λ > 0, giving a simple alternative to existing sampling algorithms for these structures. We also show rapid mixing for analogous Markov chains on dominating sets, b-edge covers, b-matchings, maximal independent sets, and maximal b-matchings. (For b-matchings, maximal independent sets, and maximal b-matchings we also require bounded degree.) Our results imply simpler alternatives to known algorithms for the sampling and approximate counting problems in these graphs. We prove our results by applying a divide-and-conquer framework we developed in a previous paper, as an alternative to the projection-restriction technique introduced by Jerrum, Son, Tetali, and Vigoda. We extend this prior framework to handle chains for which the application of that framework is not straightforward, strengthening existing results by Dyer, Goldberg, and Jerrum and by Heinrich for the Glauber dynamics on q-colorings of graphs of bounded treewidth and bounded degree.
Cite as

David Eppstein and Daniel Frishberg. Rapid Mixing for the Hardcore Glauber Dynamics and Other Markov Chains in Bounded-Treewidth Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 30:1-30:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{eppstein_et_al:LIPIcs.ISAAC.2023.30,
  author =	{Eppstein, David and Frishberg, Daniel},
  title =	{{Rapid Mixing for the Hardcore Glauber Dynamics and Other Markov Chains in Bounded-Treewidth Graphs}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{30:1--30:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.30},
  URN =		{urn:nbn:de:0030-drops-193324},
  doi =		{10.4230/LIPIcs.ISAAC.2023.30},
  annote =	{Keywords: Glauber dynamics, mixing time, projection-restriction, multicommodity flow}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.65
Range Avoidance for Constant Depth Circuits: Hardness and Algorithms

Authors: Karthik Gajulapalli, Alexander Golovnev, Satyajeet Nagargoje, and Sidhant Saraogi

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
Range Avoidance (Avoid) is a total search problem where, given a Boolean circuit 𝖢: {0,1}ⁿ → {0,1}^m, m > n, the task is to find a y ∈ {0,1}^m outside the range of 𝖢. For an integer k ≥ 2, NC⁰_k-Avoid is a special case of Avoid where each output bit of 𝖢 depends on at most k input bits. While there is a very natural randomized algorithm for Avoid, a deterministic algorithm for the problem would have many interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high formula complexity, rigid matrices, and optimal linear codes, reduce to NC⁰₄-Avoid, thus establishing conditional hardness of the NC⁰₄-Avoid problem. On the other hand, NC⁰₂-Avoid admits polynomial-time algorithms, leaving the question about the complexity of NC⁰₃-Avoid open. We give the first reduction of an explicit construction question to NC⁰₃-Avoid. Specifically, we prove that a polynomial-time algorithm (with an NP oracle) for NC⁰₃-Avoid for the case of m = n+n^{2/3} would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits. We also give deterministic polynomial-time algorithms for all NC⁰_k-Avoid problems for m ≥ n^{k-1}/log(n). Prior work required an NP oracle, and required larger stretch, m ≥ n^{k-1}.
Cite as

Karthik Gajulapalli, Alexander Golovnev, Satyajeet Nagargoje, and Sidhant Saraogi. Range Avoidance for Constant Depth Circuits: Hardness and Algorithms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 65:1-65:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gajulapalli_et_al:LIPIcs.APPROX/RANDOM.2023.65,
  author =	{Gajulapalli, Karthik and Golovnev, Alexander and Nagargoje, Satyajeet and Saraogi, Sidhant},
  title =	{{Range Avoidance for Constant Depth Circuits: Hardness and Algorithms}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{65:1--65:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.65},
  URN =		{urn:nbn:de:0030-drops-188901},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.65},
  annote =	{Keywords: Boolean function analysis, Explicit Constructions, Low-depth Circuits, Range Avoidance, Matrix Rigidity, Circuit Lower Bounds}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.21
Stability-Preserving, Time-Efficient Mechanisms for School Choice in Two Rounds

Authors: Karthik Gajulapalli, James A. Liu, Tung Mai, and Vijay V. Vazirani

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
We address the following dynamic version of the school choice question: a city, named City, admits students in two temporally-separated rounds, denoted R₁ and R₂. In round R₁, the capacity of each school is fixed and mechanism M₁ finds a student optimal stable matching. In round R₂, certain parameters change, e.g., new students move into the City or the City is happy to allocate extra seats to specific schools. We study a number of Settings of this kind and give polynomial time algorithms for obtaining a stable matching for the new situations. It is well established that switching the school of a student midway, unsynchronized with her classmates, can cause traumatic effects. This fact guides us to two types of results: the first simply disallows any re-allocations in round R₂, and the second asks for a stable matching that minimizes the number of re-allocations. For the latter, we prove that the stable matchings which minimize the number of re-allocations form a sublattice of the lattice of stable matchings. Observations about incentive compatibility are woven into these results. We also give a third type of results, namely proofs of NP-hardness for a mechanism for round R₂ under certain settings.
Cite as

Karthik Gajulapalli, James A. Liu, Tung Mai, and Vijay V. Vazirani. Stability-Preserving, Time-Efficient Mechanisms for School Choice in Two Rounds. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gajulapalli_et_al:LIPIcs.FSTTCS.2020.21,
  author =	{Gajulapalli, Karthik and Liu, James A. and Mai, Tung and Vazirani, Vijay V.},
  title =	{{Stability-Preserving, Time-Efficient Mechanisms for School Choice in Two Rounds}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.21},
  URN =		{urn:nbn:de:0030-drops-132626},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.21},
  annote =	{Keywords: stable matching, mechanism design, NP-Hardness}
}
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