161 Search Results for "Srinivasan, Srikanth"

Volume

LIPIcs, Volume 339

40th Computational Complexity Conference (CCC 2025)

CCC 2025, August 5-8, 2025, Toronto, Canada

Editors: Srikanth Srinivasan

Volume

LIPIcs, Volume 284

43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

FSTTCS 2023, December 18-20, 2023, IIIT Hyderabad, Telangana, India

Editors: Patricia Bouyer and Srikanth Srinivasan

Document
DOI: 10.4230/LIPIcs.CCC.2026.16
Constant-Depth Circuits for Polynomial GCD over Any Characteristic

Authors: Somnath Bhattacharjee, Mrinal Kumar, Shanthanu S. Rai, Varun Ramanathan, Ramprasad Saptharishi, and Shubhangi Saraf

Published in: LIPIcs, Volume 383, 41st Computational Complexity Conference (CCC 2026)

Abstract
We show that the GCD of two univariate polynomials can be computed by (piece-wise) algebraic circuits of constant depth and polynomial size over any sufficiently large field, regardless of the characteristic. This extends a recent result of Andrews & Wigderson who showed such an upper bound over fields of zero or large characteristic. Our proofs are based on a recent work of Bhattacharjee, Kumar, Rai, Ramanathan, Saptharishi & Saraf that shows closure of constant depth algebraic circuits under factorization. On our way to the proof, we show that any n-variate symmetric polynomial P that has a small constant depth algebraic circuit can be written as the composition of a small constant depth algebraic circuit with elementary symmetric polynomials. This statement is a constant depth version of a result of Bläser & Jindal, who showed this for algebraic circuits of unbounded depth. As an application of our techniques, we also strengthen the closure results for factors of constant-depth circuits in the work of Bhattacharjee et al. over fields for small characteristic.
Cite as

Somnath Bhattacharjee, Mrinal Kumar, Shanthanu S. Rai, Varun Ramanathan, Ramprasad Saptharishi, and Shubhangi Saraf. Constant-Depth Circuits for Polynomial GCD over Any Characteristic. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bhattacharjee_et_al:LIPIcs.CCC.2026.16,
  author =	{Bhattacharjee, Somnath and Kumar, Mrinal and Rai, Shanthanu S. and Ramanathan, Varun and Saptharishi, Ramprasad and Saraf, Shubhangi},
  title =	{{Constant-Depth Circuits for Polynomial GCD over Any Characteristic}},
  booktitle =	{41st Computational Complexity Conference (CCC 2026)},
  pages =	{16:1--16:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-437-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{383},
  editor =	{Moshkovitz, Dana},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2026.16},
  URN =		{urn:nbn:de:0030-drops-270580},
  doi =		{10.4230/LIPIcs.CCC.2026.16},
  annote =	{Keywords: algebraic circuits, polynomial greatest common divisor, symmetric polynomials, finite fields, constant-depth circuits}
}
Document
DOI: 10.4230/LIPIcs.CCC.2026.22
Multilinear Algebraic Branching Programs and the Min-Partition Rank Method

Authors: Théo Borém Fabris, Nutan Limaye, Srikanth Srinivasan, and Amir Yehudayoff

Published in: LIPIcs, Volume 383, 41st Computational Complexity Conference (CCC 2026)

Abstract
It is a long-standing open problem in algebraic complexity to prove lower bounds against multilinear algebraic branching programs (mlABPs), however the best lower bounds are still quadratic (Alon, Kumar and Volk (Combinatorica 2020)). At the same time, it remains a possibility that the "min-partition rank" method introduced by Raz (Theory Comput. 2006), which is used to prove all known multilinear lower bounds, can also be used to prove superpolynomial lower bounds on the size of mlABPs. In this paper, we analyze the potential of the min-partition rank method to prove lower bounds on the size of mlABPs, and show the following results: 1) We relate this method to a purely combinatorial question regarding the minimum size of set systems whose chains satisfy a discrepancy condition. In the case of set-multilinear ABPs, this combinatorial measure characterizes the best lower bound that can be achieved via the min-partition rank method. 2) We prove a non-trivial upper bound on the size of a set system satisfying this combinatorial property. Together with our construction of full-rank mlABPs from set systems, this recovers a superpolynomial separation between mlABPs and multilinear formulas (Dvir, Malod, Perifel and Yehudayoff (STOC 2012)) via a conceptually different proof. 3) The property we study extends combinatorial notions of "balancing sets" considered in previous works, for which near-tight bounds are known via intervals families. We show that any intervals set system is very far from satisfying our property. This showcases how our methods capture combinatorial structures that evade previous techniques, and also allows us to improve and generalize known lower bounds for sum of ordered set-multilinear ABPs (Chatterjee, Kush, Saraf, Shpilka (CCC 2024)). These results build a bridge between algebraic complexity theory and the behavior of random walks. Our upper bound uses the fact that, with noticeable probability, a random walk of length n on the integers returns to its starting point at least once every n/log n steps (Csáki, Erdős, and Révész (PTRF 1985)), while, for our lower bound, we prove that two independent random walks are "far" from each other in discrete Fréchet distance.
Cite as

Théo Borém Fabris, Nutan Limaye, Srikanth Srinivasan, and Amir Yehudayoff. Multilinear Algebraic Branching Programs and the Min-Partition Rank Method. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fabris_et_al:LIPIcs.CCC.2026.22,
  author =	{Fabris, Th\'{e}o Bor\'{e}m and Limaye, Nutan and Srinivasan, Srikanth and Yehudayoff, Amir},
  title =	{{Multilinear Algebraic Branching Programs and the Min-Partition Rank Method}},
  booktitle =	{41st Computational Complexity Conference (CCC 2026)},
  pages =	{22:1--22:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-437-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{383},
  editor =	{Moshkovitz, Dana},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2026.22},
  URN =		{urn:nbn:de:0030-drops-270642},
  doi =		{10.4230/LIPIcs.CCC.2026.22},
  annote =	{Keywords: Algebraic branching programs, Multilinear computations, Rank methods}
}
Document
DOI: 10.4230/LIPIcs.CCC.2026.33
Multilinear Formula Lower Bounds for Sparse Determinants

Authors: Pruthvi Boyapati, Suryajith Chillara, and Pratyush Vempati

Published in: LIPIcs, Volume 383, 41st Computational Complexity Conference (CCC 2026)

Abstract
Raz (2009) proved that multilinear formulas computing the determinant of a generic n × n matrix require size n^{Ω(log n)}. A fundamental question in understanding this lower bound is identifying which structural properties of the determinant drive this hardness. Is it the quadratic number of variables? The dense connectivity? Specific algebraic symmetries? We establish that density is not essential. We prove the existence of n × n symbolic matrices with only Θ(nlog⁶ n) nonzero entries - reducing the variable count by a factor of n/log⁶ n - such that any multilinear formula computing their determinants still requires size n^{Ω(log n)}. Our construction uses rectangle sampling from the complete bipartite graph to generate sparse matrices that simultaneously maintain perfect matchings (ensuring nonzero determinant) while exhibiting diagonal imbalance under random vertex permutations - a geometric property we identify as the key driver of factor imbalance in Raz’s framework. This demonstrates that Raz’s partial derivatives method is remarkably robust to sparsification, and suggests that the fundamental source of multilinear hardness for determinant lies in expansion-like combinatorial structure rather than density. Our techniques combine concentration inequalities for dependent random variables with insights from random graph theory.
Cite as

Pruthvi Boyapati, Suryajith Chillara, and Pratyush Vempati. Multilinear Formula Lower Bounds for Sparse Determinants. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 33:1-33:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{boyapati_et_al:LIPIcs.CCC.2026.33,
  author =	{Boyapati, Pruthvi and Chillara, Suryajith and Vempati, Pratyush},
  title =	{{Multilinear Formula Lower Bounds for Sparse Determinants}},
  booktitle =	{41st Computational Complexity Conference (CCC 2026)},
  pages =	{33:1--33:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-437-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{383},
  editor =	{Moshkovitz, Dana},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2026.33},
  URN =		{urn:nbn:de:0030-drops-270753},
  doi =		{10.4230/LIPIcs.CCC.2026.33},
  annote =	{Keywords: Determinants, Multilinear polynomials, Formula lower bounds}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.4
Unit Interval Selection in Random Order Streams

Authors: Cezar-Mihail Alexandru, Adithya Diddapur, Magnús M. Halldórsson, Christian Konrad, and Kheeran K. Naidu

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

Abstract
We consider the Unit Interval Selection problem in the one-pass random order streaming model. In this setting, an algorithm is presented with a sequence of n unit-length intervals on the line that arrive in uniform random order, one at a time, and the objective is to output (an approximation of) a largest set of disjoint intervals using space linear in the size of an optimal solution. Previous work only considered adversarially ordered streams and established that, within these space constraints, a (2/3)-approximation can be achieved in such streams, and this is best possible, in that going beyond such an approximation factor requires space Ω(n) [Emek et al., TALG'16]. In this work, we show that an improved expected approximation factor can be achieved if the input stream is in uniform random order, where the expectation is taken over the stream order. More specifically, we give a one-pass streaming algorithm with expected approximation factor 0.7401 that uses space O(|OPT|), where OPT denotes an optimal solution. We also show that random order algorithms with expected approximation factor above 8/9 require space Ω(n), and algorithms that compute a better than 2/3-approximation with probability above 2/3 also require Ω(n) space. On a technical level, we design an algorithm for the restricted domain [0, Δ), for some constant Δ, and use standard techniques to obtain an algorithm for unrestricted domains. For the restricted domain [0, Δ), we run O(Δ) recursive instances of our algorithm, with each instance targeting the situation where a specific interval of an optimal solution arrives first. We establish the interesting property of our algorithm that it performs worst when the input stream consists solely of a set of independent intervals. It then remains to analyse the algorithm on these simple instances. Our lower bound is proved via communication complexity arguments, similar in spirit to the robust communication lower bounds established by [Chakrabarti et al., Theory Comput. 2016].
Cite as

Cezar-Mihail Alexandru, Adithya Diddapur, Magnús M. Halldórsson, Christian Konrad, and Kheeran K. Naidu. Unit Interval Selection in Random Order Streams. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{alexandru_et_al:LIPIcs.STACS.2026.4,
  author =	{Alexandru, Cezar-Mihail and Diddapur, Adithya and Halld\'{o}rsson, Magn\'{u}s M. and Konrad, Christian and Naidu, Kheeran K.},
  title =	{{Unit Interval Selection in Random Order Streams}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{4:1--4:20},
  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.2026.4},
  URN =		{urn:nbn:de:0030-drops-254933},
  doi =		{10.4230/LIPIcs.STACS.2026.4},
  annote =	{Keywords: Random order streaming algorithms, unit interval selection}
}
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.2026.112
Lower Bounds Beyond DNF of Parities

Authors: Artur Riazanov, Anastasia Sofronova, and Dmitry Sokolov

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

Abstract
We consider a subclass of AC⁰[2] circuits that simultaneously captures DNF∘Xor and depth-3 AC⁰ circuits. For this class we show a technique for proving lower bounds inspired by the top-down approach. We give lower bounds for the middle slice function, inner product function, and affine dispersers.
Cite as

Artur Riazanov, Anastasia Sofronova, and Dmitry Sokolov. Lower Bounds Beyond DNF of Parities. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 112:1-112:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{riazanov_et_al:LIPIcs.ITCS.2026.112,
  author =	{Riazanov, Artur and Sofronova, Anastasia and Sokolov, Dmitry},
  title =	{{Lower Bounds Beyond DNF of Parities}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{112:1--112:15},
  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.2026.112},
  URN =		{urn:nbn:de:0030-drops-253996},
  doi =		{10.4230/LIPIcs.ITCS.2026.112},
  annote =	{Keywords: boolean circuits, top-down, unpredictability}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.49
Debordering Closure Results in Determinantal and Pfaffian Ideals

Authors: Anakin Dey and Zeyu Guo

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

Abstract
One important question in algebraic complexity is understanding the complexity of polynomial ideals (Grochow, Bulletin of EATCS 131, 2020). Andrews and Forbes (STOC 2022) studied the determinantal ideals I^{det}_{n,m,r} generated by the r× r minors of n× m matrices. Over fields of characteristic zero or of sufficiently large characteristic, they showed that for any nonzero f ∈ I^{det}_{n,m,r}, the determinant of a t × t matrix of variables with t = Θ{r^{1/3}} is approximately computed by a constant-depth, polynomial-size f-oracle algebraic circuit, in the sense that the determinant lies in the border of such circuits. An analogous result was also obtained for Pfaffians in the same paper. In this work, we deborder the result of Andrews and Forbes by showing that when f has polynomial degree, the determinant is in fact exactly computed by a constant-depth, polynomial-size f-oracle algebraic circuit. We further establish an analogous result for Pfaffian ideals. Our results are established using the isolation lemma, combined with a careful analysis of straightening-law expansions of polynomials in determinantal and Pfaffian ideals.
Cite as

Anakin Dey and Zeyu Guo. Debordering Closure Results in Determinantal and Pfaffian Ideals. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 49:1-49:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dey_et_al:LIPIcs.ITCS.2026.49,
  author =	{Dey, Anakin and Guo, Zeyu},
  title =	{{Debordering Closure Results in Determinantal and Pfaffian Ideals}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{49:1--49:24},
  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.2026.49},
  URN =		{urn:nbn:de:0030-drops-253363},
  doi =		{10.4230/LIPIcs.ITCS.2026.49},
  annote =	{Keywords: Algebraic circuit complexity, Isolation lemma, Debordering}
}
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.2026.70
Unconditional Pseudorandomness Against Shallow Quantum Circuits

Authors: Soumik Ghosh, Sathyawageeswar Subramanian, and Wei Zhan

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

Abstract
Quantum computational pseudorandomness has emerged as a fundamental notion that spans connections to complexity theory, cryptography and fundamental physics. However, all known constructions of efficient quantum-secure pseudorandom objects rely on complexity theoretic assumptions. In this work, we establish the first unconditionally secure efficient pseudorandom constructions against shallow-depth quantum circuit classes. We prove that: - Any quantum state 2-design yields unconditional pseudorandomness against both QNC⁰ circuits with arbitrarily many ancillae and AC⁰∘QNC⁰ circuits with nearly linear ancillae. - Random phased subspace states, where the phases are picked using a 4-wise independent function, are unconditionally pseudoentangled against the above circuit classes. - Any unitary 2-design yields unconditionally secure parallel-query pseudorandom unitaries against geometrically local QNC⁰ adversaries, even with limited AC⁰ postprocessing. Our results stand in stark contrast to the standard guarantee of the 2-design property, which only ensures that they cannot be distinguished from Haar random ensembles using two copies or queries. Our work demonstrates that quantum computational pseudorandomness can be achieved unconditionally for natural classes of restricted adversaries, opening new directions in quantum complexity theory.
Cite as

Soumik Ghosh, Sathyawageeswar Subramanian, and Wei Zhan. Unconditional Pseudorandomness Against Shallow Quantum Circuits. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 70:1-70:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ghosh_et_al:LIPIcs.ITCS.2026.70,
  author =	{Ghosh, Soumik and Subramanian, Sathyawageeswar and Zhan, Wei},
  title =	{{Unconditional Pseudorandomness Against Shallow Quantum Circuits}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{70:1--70:25},
  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.2026.70},
  URN =		{urn:nbn:de:0030-drops-253578},
  doi =		{10.4230/LIPIcs.ITCS.2026.70},
  annote =	{Keywords: quantum pseudorandomness, shallow quantum circuits, pseudorandomness, t-designs}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.113
Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP

Authors: Benjamin Rossman and Davidson Zhu

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

Abstract
The sum-of-squares (SoS) complexity of a d-multiquadratic polynomial f (quadratic in each of d blocks of n variables) is the minimum s such that f = ∑_{i = 1}^s g_i² with each g_i d-multilinear. In the case d = 2, Hrubeš, Wigderson and Yehudayoff [Hrubeš et al., 2011] showed that an n^{1+Ω(1)} lower bound on the SoS complexity of explicit biquadratic polynomials implies an exponential lower bound for non-commutative arithmetic circuits. In this paper, we establish an analogous connection between general multiquadratic sum-of-squares and commutative arithmetic formulas. Specifically, we show that an n^{d-o(log d)} lower bound on the SoS complexity of explicit d-multiquadratic polynomials, for any d = d(n) with ω(1) ≤ d(n) ≤ O((log n)/(log log n)), would separate the algebraic complexity classes VNC¹ and VNP.
Cite as

Benjamin Rossman and Davidson Zhu. Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 113:1-113:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{rossman_et_al:LIPIcs.ITCS.2026.113,
  author =	{Rossman, Benjamin and Zhu, Davidson},
  title =	{{Multi-Quadratic Sum-Of-Squares Lower Bounds Imply VNC ¹ ≠ VNP}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{113:1--113:22},
  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.2026.113},
  URN =		{urn:nbn:de:0030-drops-254006},
  doi =		{10.4230/LIPIcs.ITCS.2026.113},
  annote =	{Keywords: sum-of-squares, arithmetic formulas}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.28
Linear Time Encodable Binary Code Achieving GV Bound with Linear Time Encodable Dual Achieving GV Bound

Authors: Martijn Brehm and Nicolas Resch

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

Abstract
We initiate the study of what we term "fast good codes" with "fast good duals." Specifically, we consider the task of constructing a binary linear code C ≤ 𝔽₂ⁿ such that both it and its dual C^⟂ : = {x ∈ 𝔽₂ⁿ:∀ c ∈ C, ⟨ x,c⟩ = 0} are asymptotically good (in fact, have rate-distance tradeoff approaching the GV bound), and are encodable in O(n) time. While we believe such codes should find applications more broadly, as motivation we describe how such codes can be used the secure computation task of encrypted matrix-vector product, as studied by Behhamouda et al (CCS 2025). Our main contribution is a construction of such a fast good code with fast good dual. Our construction is inspired by the repeat multiple accumulate (RMA) codes of Divsalar, Jin and McEliece (Allerton, 1998). To create the rate 1/2 code, after repeating each message coordinate, we perform accumulation steps - where first a uniform coordinate permutation is applied, and afterwards the prefix-sum modulo 2 is applied - which are alternated with discrete derivative steps - where again a uniform coordinate permutation is applied, and afterwards the previous two coordinates are summed modulo 2. Importantly, these two operations are inverse of each other. In particular, the dual of the code is very similar, with the accumulation and discrete derivative steps reversed. Our analysis is inspired by a prior analysis of RMA codes due to Ravazzi and Fagnani (IEEE Trans. Info. Theory, 2009). The main idea is to bound the input-output weight-enumerator function: the expected number of messages of a given weight that are encoded into a codeword of a given weight. We face new challenges in controlling the behaviour of the discrete derivative matrix (which can significantly drop the weight of a vector), which we overcome by careful case analysis.
Cite as

Martijn Brehm and Nicolas Resch. Linear Time Encodable Binary Code Achieving GV Bound with Linear Time Encodable Dual Achieving GV Bound. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 28:1-28:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brehm_et_al:LIPIcs.ITCS.2026.28,
  author =	{Brehm, Martijn and Resch, Nicolas},
  title =	{{Linear Time Encodable Binary Code Achieving GV Bound with Linear Time Encodable Dual Achieving GV Bound}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{28:1--28:19},
  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.2026.28},
  URN =		{urn:nbn:de:0030-drops-253157},
  doi =		{10.4230/LIPIcs.ITCS.2026.28},
  annote =	{Keywords: Binary error-correcting codes, dual codes, fast encoding, repeat-multiple-accumulate codes}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.99
AC⁰[p]-Frege Cannot Efficiently Prove That Constant-Depth Algebraic Circuit Lower Bounds Are Hard

Authors: Jiaqi Lu, Rahul Santhanam, and Iddo Tzameret

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

Abstract
We study whether lower bounds against constant-depth algebraic circuits computing the Permanent over finite fields (Limaye-Srinivasan-Tavenas [J. ACM, 2025] and Forbes [CCC'24]) are hard to prove in certain proof systems. We focus on a DNF formula that expresses that such lower bounds are hard for constant-depth algebraic proofs. Using an adaptation of the diagonalization framework of Santhanam and Tzameret (SIAM J. Comput., 2025), we show unconditionally that this family of DNF formulas does not admit polynomial-size propositional AC⁰[p]-Frege proofs, infinitely often. This rules out the possibility that the DNF family is easy, and establishes that its status is either that of a hard tautology for AC⁰[p]-Frege or else unprovable (i.e., not a tautology). While it remains open whether the DNFs in question are tautologies, we provide evidence in this direction. In particular, under the plausible assumption that certain (weak) properties of multilinear algebra - specifically, those involving tensor rank - do not admit short constant-depth algebraic proofs, the DNFs are tautologies. We also observe that several weaker variants of the DNF formula are provably tautologies, and we show that the question of whether the DNFs are tautologies connects to conjectures of Razborov (ICALP'96) and Krajíček (J. Symb. Log., 2004). Additionally, our result has the following special features: ii) Existential depth amplification: the DNF formula considered is parameterised by a constant depth d bounding the depth of the algebraic proofs. We show that there exists some fixed depth d such that if there are no small depth-d algebraic proofs of certain circuit lower bounds for the Permanent, then there are no such small algebraic proofs in any constant depth. iii) Necessity: We show that our result is a necessary step towards establishing lower bounds against constant-depth algebraic proofs, and more generally against any sufficiently strong proof system. In particular, showing there are no short proofs for our DNF formulas, obtained by replacing "constant-depth algebraic circuits" with any "reasonable" algebraic circuit class C, is necessary in order to prove any super-polynomial lower bounds against algebraic proofs operating with circuits from C.
Cite as

Jiaqi Lu, Rahul Santhanam, and Iddo Tzameret. AC⁰[p]-Frege Cannot Efficiently Prove That Constant-Depth Algebraic Circuit Lower Bounds Are Hard. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 99:1-99:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{lu_et_al:LIPIcs.ITCS.2026.99,
  author =	{Lu, Jiaqi and Santhanam, Rahul and Tzameret, Iddo},
  title =	{{AC⁰\lbrackp\rbrack-Frege Cannot Efficiently Prove That Constant-Depth Algebraic Circuit Lower Bounds Are Hard}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{99:1--99:25},
  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.2026.99},
  URN =		{urn:nbn:de:0030-drops-253865},
  doi =		{10.4230/LIPIcs.ITCS.2026.99},
  annote =	{Keywords: Complexity, Lower bounds, Proof complexity, AC⁰\lbrackp\rbrack-Frege, Diagonalisation, Algebraic complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.9
On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

Authors: Robert Andrews, Jules Armand, Prateek Dwivedi, Magnus Rahbek Dalgaard Hansen, Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas

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

Abstract
We investigate the closure properties of read-once oblivious Algebraic Branching Programs (roABPs) under various natural algebraic operations and prove the following. - Non-closure under factoring: There is a sequence of explicit polynomials (f_n(x₁,…, x_n))_n that have poly(n)-sized roABPs such that some irreducible factor of f_n requires roABPs of superpolynomial size in any order. - Non-closure under powering: There is a sequence of polynomials (f_n(x₁,…, x_n))_n with poly(n)-sized roABPs such that any super-constant power of f_n does not have roABPs of polynomial size in any order (and f_nⁿ requires exponential size in any order). - Non-closure under symmetric operations: There are symmetric polynomials (f_n(e₁,…, e_n))_n that have roABPs of polynomial size such that f_n(x₁,…, x_n) do not have roABPs of subexponential size. (Here, e₁,…, e_n denote the elementary symmetric polynomials in n variables.) These results should be viewed in light of known results on models such as algebraic circuits, (general) algebraic branching programs, formulas and constant-depth circuits, all of which are known to be closed under these operations. To prove non-closure under factoring, we construct hard polynomials based on expander graphs using gadgets that lift their hardness from sparse polynomials to roABPs. For symmetric compositions, we show that the circulant polynomial requires roABPs of exponential size in every variable order.
Cite as

Robert Andrews, Jules Armand, Prateek Dwivedi, Magnus Rahbek Dalgaard Hansen, Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. On Closure Properties of Read-Once Oblivious Algebraic Branching Programs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{andrews_et_al:LIPIcs.ITCS.2026.9,
  author =	{Andrews, Robert and Armand, Jules and Dwivedi, Prateek and Hansen, Magnus Rahbek Dalgaard and Limaye, Nutan and Srinivasan, Srikanth and Tavenas, S\'{e}bastien},
  title =	{{On Closure Properties of Read-Once Oblivious Algebraic Branching Programs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{9:1--9:21},
  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.2026.9},
  URN =		{urn:nbn:de:0030-drops-252964},
  doi =		{10.4230/LIPIcs.ITCS.2026.9},
  annote =	{Keywords: Factoring, Closure Properties, Sparsity Bounds, Symmetric Polynomials, roABP, Expander Graphs}
}
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