5 Search Results for "Liang, Jingxun"

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.2026.37
New Algebrization Barriers to Circuit Lower Bounds via Communication Complexity of Missing-String

Authors: Lijie Chen, Yang Hu, and Hanlin Ren

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

Abstract
The algebrization barrier, proposed by Aaronson and Wigderson (STOC '08, ToCT '09), captures the limitations of many complexity-theoretic techniques based on arithmetization. Notably, several circuit lower bounds that overcome the relativization barrier (Buhrman-Fortnow-Thierauf, CCC '98; Vinodchandran, TCS '05; Santhanam, STOC '07, SICOMP '09) remain subject to the algebrization barrier. In this work, we establish several new algebrization barriers to circuit lower bounds by studying the communication complexity of the following problem, called XOR-Missing-String: For m < 2^{n/2}, Alice gets a list of m strings x₁, … , x_m ∈ {0, 1}ⁿ, Bob gets a list of m strings y₁, … , y_m ∈ {0, 1}ⁿ, and the goal is to output a string s ∈ {0, 1}ⁿ that is not equal to x_i⊕ y_j for any i, j ∈ [m]. 1) We construct an oracle A₁ and its multilinear extension A₁̃ such that PostBPE^{A₁̃} has linear-size A₁-oracle circuits on infinitely many input lengths. That is, proving PostBPE ̸ ⊆ i.o.- SIZE[O(n)] requires non-algebrizing techniques. This barrier follows from a PostBPP communication lower bound for XOR-Missing-String. This is in contrast to the well-known algebrizing lower bound MA_E (⊆ PostBPE) ̸ ⊆ P/_poly. 2) We construct an oracle A₂ and its multilinear extension A₂̃ such that BPE^{A₂̃} has linear-size A₂-oracle circuits on all input lengths. Previously, a similar barrier was demonstrated by Aaronson and Wigderson, but in their result, A₂̃ is only a multiquadratic extension of A₂. Our results show that communication complexity is more useful than previously thought for proving algebrization barriers, as Aaronson and Wigderson wrote that communication-based barriers were "more contrived". This serves as an example of how XOR-Missing-String forms new connections between communication lower bounds and algebrization barriers. 3) Finally, we study algebrization barriers to circuit lower bounds for MA_E. Buhrman, Fortnow, and Thierauf proved a sub-half-exponential circuit lower bound for MA_E via algebrizing techniques. Toward understanding whether the half-exponential bound can be improved, we define a natural subclass of MA_E that includes their hard MA_E language, and prove the following result: For every super-half-exponential function h(n), we construct an oracle A₃ and its multilinear extension A₃̃ such that this natural subclass of MA_E^{A₃̃} has h(n)-size A₃-oracle circuits on all input lengths. This suggests that half-exponential might be the correct barrier for MA_E circuit lower bounds w.r.t. algebrizing techniques.
Cite as

Lijie Chen, Yang Hu, and Hanlin Ren. New Algebrization Barriers to Circuit Lower Bounds via Communication Complexity of Missing-String. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2026.37,
  author =	{Chen, Lijie and Hu, Yang and Ren, Hanlin},
  title =	{{New Algebrization Barriers to Circuit Lower Bounds via Communication Complexity of Missing-String}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{37:1--37:20},
  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.2026.37},
  URN =		{urn:nbn:de:0030-drops-253246},
  doi =		{10.4230/LIPIcs.ITCS.2026.37},
  annote =	{Keywords: circuit lower bound, algebrization barrier, missing string, communication complexity}
}
Document
DOI: 10.4230/LIPIcs.SEA.2025.8
A Simple Integer Successor-Delete Data Structure

Authors: Gerth Stølting Brodal

Published in: LIPIcs, Volume 338, 23rd International Symposium on Experimental Algorithms (SEA 2025)

Abstract
We consider a simple decremental data structure for maintaining a set of integers, that supports initializing the set to {1,2,…,n} followed by d deletions and s successor queries in arbitrary order in total 𝒪(n+d+s⋅(1+log_{max(2,s/n)} min(s,n))) time. The data structure consists of a single array of n integers. A straightforward modification allows the data structure to also support p predecessor and r range queries, with a total output k, in total 𝒪(n+d+k+q ⋅ (1+log_{max(2,q/n)} min(q,n))) time, where q = s+p+r. The data structure is essentially a special case of the classic union-find data structure with path compression but with unweighted linking (i.e., without linking by rank or size), that is known to achieve logarithmic amortized time bounds (Tarjan and van Leeuwen, 1984). In this paper we study the efficiency of this simple data structure, and compare it to other, theoretically superior, data structures.
Cite as

Gerth Stølting Brodal. A Simple Integer Successor-Delete Data Structure. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{brodal:LIPIcs.SEA.2025.8,
  author =	{Brodal, Gerth St{\o}lting},
  title =	{{A Simple Integer Successor-Delete Data Structure}},
  booktitle =	{23rd International Symposium on Experimental Algorithms (SEA 2025)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-375-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{338},
  editor =	{Mutzel, Petra and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2025.8},
  URN =		{urn:nbn:de:0030-drops-232461},
  doi =		{10.4230/LIPIcs.SEA.2025.8},
  annote =	{Keywords: Successor queries, deletions, interval union-find, union-find}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.114
Optimal Static Fully Indexable Dictionaries

Authors: Jingxun Liang and Renfei Zhou

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Fully indexable dictionaries (FID) store sets of integer keys while supporting rank/select queries. They serve as basic building blocks in many succinct data structures. Despite the great importance of FIDs, no known FID is succinct with efficient query time when the universe size U is a large polynomial in the number of keys n, which is the conventional parameter regime for dictionary problems. In this paper, we design an FID that uses log binom(U,n) + n/((log U/t)^{Ω(t)}) bits of space, and answers rank/select queries in O(t + log log n) time in the worst case, for any parameter 1 ≤ t ≤ log n / log log n, provided U = n^{1 + Θ(1)}. This time-space trade-off matches known lower bounds for FIDs [Pǎtraşcu and Thorup, 2006; Pǎtraşcu and Viola, 2010; Viola, 2023] when t ≤ log^{0.99} n. Our techniques also lead to efficient succinct data structures for the fundamental problem of maintaining n integers each of 𝓁 = Θ(log n) bits and supporting partial-sum queries, with a trade-off between O(t) query time and n𝓁 + n / (log n / t)^{Ω(t)} bits of space. Prior to this work, no known data structure for the partial-sum problem achieves constant query time with n 𝓁 + o(n) bits of space usage.
Cite as

Jingxun Liang and Renfei Zhou. Optimal Static Fully Indexable Dictionaries. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 114:1-114:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{liang_et_al:LIPIcs.ICALP.2025.114,
  author =	{Liang, Jingxun and Zhou, Renfei},
  title =	{{Optimal Static Fully Indexable Dictionaries}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{114:1--114:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.114},
  URN =		{urn:nbn:de:0030-drops-234918},
  doi =		{10.4230/LIPIcs.ICALP.2025.114},
  annote =	{Keywords: data structures, dictionaries, space efficiency}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.80
On the Perturbation Function of Ranking and Balance for Weighted Online Bipartite Matching

Authors: Jingxun Liang, Zhihao Gavin Tang, Yixuan Even Xu, Yuhao Zhang, and Renfei Zhou

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
Ranking and Balance are arguably the two most important algorithms in the online matching literature. They achieve the same optimal competitive ratio of 1-1/e for the integral version and fractional version of online bipartite matching by Karp, Vazirani, and Vazirani (STOC 1990) respectively. The two algorithms have been generalized to weighted online bipartite matching problems, including vertex-weighted online bipartite matching and AdWords, by utilizing a perturbation function. The canonical choice of the perturbation function is f(x) = 1-e^{x-1} as it leads to the optimal competitive ratio of 1-1/e in both settings. We advance the understanding of the weighted generalizations of Ranking and Balance in this paper, with a focus on studying the effect of different perturbation functions. First, we prove that the canonical perturbation function is the unique optimal perturbation function for vertex-weighted online bipartite matching. In stark contrast, all perturbation functions achieve the optimal competitive ratio of 1-1/e in the unweighted setting. Second, we prove that the generalization of Ranking to AdWords with unknown budgets using the canonical perturbation function is at most 0.624 competitive, refuting a conjecture of Vazirani (2021). More generally, as an application of the first result, we prove that no perturbation function leads to the prominent competitive ratio of 1-1/e by establishing an upper bound of 1-1/e-0.0003. Finally, we propose the online budget-additive welfare maximization problem that is intermediate between AdWords and AdWords with unknown budgets, and we design an optimal 1-1/e competitive algorithm by generalizing Balance.
Cite as

Jingxun Liang, Zhihao Gavin Tang, Yixuan Even Xu, Yuhao Zhang, and Renfei Zhou. On the Perturbation Function of Ranking and Balance for Weighted Online Bipartite Matching. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 80:1-80:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{liang_et_al:LIPIcs.ESA.2023.80,
  author =	{Liang, Jingxun and Tang, Zhihao Gavin and Xu, Yixuan Even and Zhang, Yuhao and Zhou, Renfei},
  title =	{{On the Perturbation Function of Ranking and Balance for Weighted Online Bipartite Matching}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{80:1--80:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.80},
  URN =		{urn:nbn:de:0030-drops-187334},
  doi =		{10.4230/LIPIcs.ESA.2023.80},
  annote =	{Keywords: Online Matching, AdWords, Ranking, Water-Filling}
}
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