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LIPIcs, Volume 383
41st Computational Complexity Conference (CCC 2026)
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Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Computational Complexity Conference (CCC)
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- published at: 2026-07-23
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-437-6
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Complete Volume
LIPIcs, Volume 383, CCC 2026, Complete Volume
Abstract
LIPIcs, Volume 383, CCC 2026, Complete Volume
Cite as
41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 1-1054, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{moshkovitz:LIPIcs.CCC.2026,
title = {{LIPIcs, Volume 383, CCC 2026, Complete Volume}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {1--1054},
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},
URN = {urn:nbn:de:0030-drops-273215},
doi = {10.4230/LIPIcs.CCC.2026},
annote = {Keywords: LIPIcs, Volume 383, CCC 2026, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{moshkovitz:LIPIcs.CCC.2026.0,
author = {Moshkovitz, Dana},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {0:i--0:xviii},
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.2026.0},
URN = {urn:nbn:de:0030-drops-273205},
doi = {10.4230/LIPIcs.CCC.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding Under ETH
Abstract
We present a simple deterministic reduction which, assuming the Exponential Time Hypothesis (ETH), yields tight lower bounds for approximating the parameterized Maximum Likelihood Decoding problem (MLD) and the parameterized Nearest Codeword Problem (NCP) within some fixed constant factor. Our starting point is the ETH-based exponential-time hardness of (c, s)-Gap MAXLIN established in [Nir Bitansky et al., 2024]. We transform a (c, s)-Gap MAXLIN instance into an instance of γ-Gap k-MLD via a novel combinatorial object that we call a cover family. We provide both a randomized construction of the required cover families and a subsequent derandomization. Prior to our work, n^{Ω(k)} hardness for constant-factor approximation was only shown under the randomized Gap Exponential Time Hypothesis Gap-ETH [Pasin Manurangsi, 2020], which is a much stronger assumption than ETH. Under ETH, the strongest known lower bound was n^{Ω(k/poly log k)} due to [Mitali Bafna et al., 2025]. Unlike previous approaches that rely on reductions from the hardness of approximating 2-CSP, our reduction provides a more direct and conceptually simpler route to achieving the optimal lower bounds.
Cite as
Rishav Gupta, Bingkai Lin, and Xin Zheng. Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding Under ETH. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 1:1-1:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gupta_et_al:LIPIcs.CCC.2026.1,
author = {Gupta, Rishav and Lin, Bingkai and Zheng, Xin},
title = {{Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding Under ETH}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {1:1--1:17},
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.2026.1},
URN = {urn:nbn:de:0030-drops-270439},
doi = {10.4230/LIPIcs.CCC.2026.1},
annote = {Keywords: Maximum Likelihood Decoding, Parameterized Complexity, Hardness of Approximation, Exponential Time Hypothesis}
}
Document
Bounded-Independence Sampling of Edges for Combinatorial Graph Properties
Abstract
Random subsampling of edges is a commonly employed technique in graph algorithms, underlying a vast array of modern algorithmic breakthroughs. Unfortunately, using this technique often leads to randomized algorithms with no clear path to derandomization because the analyses rely on a union bound over exponentially many events. In this work, we revisit this goal of derandomizing randomized sampling in graphs.
We give several results related to bounded-independence edge subsampling, and in the process of doing so, generalize several of the results of Alon and Nussboim (FOCS 2008), who studied bounded-independence analogues of random graphs (which can be viewed as edge subsamples of the complete graph). Most notably, we show:
1) O(log(m))-wise independence suffices for preserving connectivity when sampling at rate 1/2 in a graph with minimum cut ≥ κ log(m) with probability 1 - 1/poly(m) (for a sufficiently large constant κ).
2) O(log(m))-wise (1/poly(m))-almost independence suffices for ensuring cycle-freeness when sampling at rate 1/2 in a graph with minimum cycle length ≥ κ log(m) with probability 1 - 1/poly(m) (for a sufficiently large constant κ).
3) If we relax to arbitrary distributions, we show there is an explicit distribution with marginals ≤ 1/2 generated using O(log(m)log log(m)) random bits such that in a graph with minimum cut ≥ κ log(m) (for a sufficiently large constant κ), a sample from the distribution has is still connected with probability 1- 1/poly(m).
To demonstrate the utility of our results, we revisit the classic problem of using parallel algorithms to find graphic matroid bases, first studied in the work of Karp, Upfal, and Wigderson (FOCS 1985). In this regime, we show that the optimal algorithms of Khanna, Putterman, and Song (arxiv 2025) can be explicitly derandomized while maintaining near-optimality.
Cite as
Aaron Putterman, Salil Vadhan, and Vadim Zaripov. Bounded-Independence Sampling of Edges for Combinatorial Graph Properties. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 2:1-2:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{putterman_et_al:LIPIcs.CCC.2026.2,
author = {Putterman, Aaron and Vadhan, Salil and Zaripov, Vadim},
title = {{Bounded-Independence Sampling of Edges for Combinatorial Graph Properties}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {2:1--2:22},
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.2026.2},
URN = {urn:nbn:de:0030-drops-270444},
doi = {10.4230/LIPIcs.CCC.2026.2},
annote = {Keywords: Graphs, random sampling}
}
Document
Improved Bounds on the Space Complexity of Circuit Evaluation
Abstract
Williams (STOC 2025) recently proved that time-t multitape Turing machines can be simulated using O(√{t log t}) space using the Cook-Mertz (STOC 2024) tree evaluation procedure. As Williams notes, applying this result to fast algorithms for the circuit value problem implies an O(√s ⋅ polylog s) space algorithm for evaluating circuits with s gates.
In this work, we provide a direct reduction from circuit value to tree evaluation without passing through Turing machines, simultaneously improving the bound to O(√{s log s}) space and providing a proof with fewer layers of abstraction.
This result can be thought of as a "sibling" result to Williams' for circuit complexity instead of time; in particular, using the fact that time-t Turing machines have size O(t log t) circuits, we can recover a slightly weakened version of Williams' result, simulating time-t machines in space O(√t log t).
Cite as
Yakov Shalunov. Improved Bounds on the Space Complexity of Circuit Evaluation. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 3:1-3:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{shalunov:LIPIcs.CCC.2026.3,
author = {Shalunov, Yakov},
title = {{Improved Bounds on the Space Complexity of Circuit Evaluation}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {3:1--3:13},
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.2026.3},
URN = {urn:nbn:de:0030-drops-270451},
doi = {10.4230/LIPIcs.CCC.2026.3},
annote = {Keywords: circuit value problem CVP, space complexity, tree evaluation problem}
}
Document
Probabilistically Checking Quantum Proofs, with Interaction
Abstract
The model of interactive oracle proofs (IOP) generalizes the notion of probabilistically checkable proof (PCP), in which a static proof is verified probabilistically by querying a small number of bits, to the interactive setting: a polynomial-time verifier interacts with an unbounded prover, but is restricted to only reading a small number of bits, in total, from the messages sent by the prover. IOPs provide a relaxed setting in which to study local probabilistic verification. They have proved instrumental in devising efficient methods for verification through subsequent compilation into non-interactive or succinct protoocls.
We study a quantum analogue of interactive oracle proofs (qIOP) in which the verifier and communication are both allowed to be quantum; yet the verifier is restricted to perform measurements only on a small number of qubits received from the prover. Our main result is a qIOP for any language in QMA, in which the total communication is polynomial but the verifier only reads a polylogarithmic number of qubits in total. The protocol has completeness parameter exponentially close to 1 and soundness bounded away from 1 by a constant. In the absence of a quantum PCP theorem, this provides the first information-theoretically sound local and robust characterization of QMA, albeit interactive. Previous works in the information-theoretic setting either considered two isolated but entangled quantum provers or quantum verifiers whose effort in a single round is small but remains polynomial when aggregated across all rounds of the protocol.
Our protocol combines the use of a quantum locally testable code (LTC) with classical techniques, notably probabilistically checkable proofs of proximity (PCPP). We avoid the necessity for complex multi-qubit tests employed in other settings by leveraging the local indistinguishability property of the quantum LTC.
Cite as
Baocheng Sun and Thomas Vidick. Probabilistically Checking Quantum Proofs, with Interaction. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 4:1-4:49, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{sun_et_al:LIPIcs.CCC.2026.4,
author = {Sun, Baocheng and Vidick, Thomas},
title = {{Probabilistically Checking Quantum Proofs, with Interaction}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {4:1--4:49},
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.2026.4},
URN = {urn:nbn:de:0030-drops-270463},
doi = {10.4230/LIPIcs.CCC.2026.4},
annote = {Keywords: quantum complexity theory, quantum probabilistically checkable proofs, interactive oracle proofs, quantum locally testable codes, QMA}
}
Document
Condensing and Extracting Against Online Adversaries
Abstract
We investigate the tasks of deterministically condensing and extracting randomness from Online Non-Oblivious Symbol Fixing (oNOSF) sources, a natural model of defective random sources for which it is known that extraction is impossible in many parameter regimes [AORSV, EUROCRYPT'20]. A (g,𝓁)-oNOSF source is a sequence of 𝓁 blocks 𝐗 = (𝐗₁, … , 𝐗_{𝓁})∼ ({0, 1}ⁿ)^{𝓁}, where at least g of the blocks are good (are independent and have some min-entropy), and the remaining bad blocks are controlled by an online adversary where each bad block can be arbitrarily correlated with any block that appears before it.
The existence of condensers (in regimes where extraction is impossible) was recently studied in [CGR, FOCS'24]. They proved condensing impossibility results for various values of g and 𝓁, and they showed the existence of condensers matching the impossibility results in the special case when n is exponential in 𝓁 (i.e., the setting of few blocks of large length).
In this work, not only do we construct the first explicit condensers matching the existential results of [CGR, FOCS'24], but we make a doubly exponential improvement by handling the case when n is only polylogarithmic in 𝓁. We also obtain a much improved explicit construction for transforming low-entropy oNOSF sources (where the good blocks only have min-entropy, as opposed to being uniform) into uniform oNOSF sources.
As our next result, we essentially resolve the question of the existence of condensers for oNOSF sources by showing the existence of condensers in almost all parameter regimes, even when n is a large enough constant and 𝓁 is growing.
We find interesting connections and applications of our results on condensers to collective coin flipping and collective sampling, problems that are well-studied in fault-tolerant distributed computing. We use our condensers to provide very simple protocols for these problems.
Next, we turn to understanding the possibility of extraction from oNOSF sources. For proving lower bounds, we introduce and initiate a systematic study of a new, natural notion of the influence of functions, which we call online influence, and establish tight bounds on the total online influence of functions, which imply extraction lower bounds. Lastly, we give explicit extractor constructions for oNOSF sources using novel connections to leader election protocols, and we further construct the required leader election protocols. These extractor constructions achieve parameters that go beyond the standard resilient functions of [AL, Combinatorica'93].
Cite as
Eshan Chattopadhyay, Mohit Gurumukhani, Noam Ringach, and Rocco A. Servedio. Condensing and Extracting Against Online Adversaries. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2026.5,
author = {Chattopadhyay, Eshan and Gurumukhani, Mohit and Ringach, Noam and Servedio, Rocco A.},
title = {{Condensing and Extracting Against Online Adversaries}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {5:1--5:22},
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.2026.5},
URN = {urn:nbn:de:0030-drops-270477},
doi = {10.4230/LIPIcs.CCC.2026.5},
annote = {Keywords: collective coin flipping, leader election, Boolean function analysis, fault tolerant distributed computing, full information model, resilient function, pseudorandomness, condensers, adversarial sources, non-oblivious symbol fixing sources, Chor-Goldreich sources}
}
Document
Improved Parallel Repetition for GHZ-Supported Games via Spreadness
Abstract
We prove that for any 3-player game G, whose query distribution has the same support as the GHZ game (i.e., all x,y,z ∈ {0,1} satisfying x+y+z = 0 (mod 2)), the value of the n-fold parallel repetition of G decays exponentially fast:
val(G^{⊗ n}) ≤ exp(-n^c)
for all sufficiently large n, where c > 0 is an absolute constant.
We also prove a concentration bound for the parallel repetition of the GHZ game: For any constant ε > 0, the probability that the players win at least a (3/4+ε) fraction of the n coordinates is at most exp(-n^c), where c = c(ε) > 0 is a constant.
In both settings, our work exponentially improves upon the previous best known bounds which were only polynomially small, i.e., of the order n^{-Ω(1)}. Our key technical tool is the notion of algebraic spreadness adapted from the breakthrough work of Kelley and Meka (FOCS '23) on sets free of 3-term progressions.
Cite as
Yang P. Liu, Shachar Lovett, and Kunal Mittal. Improved Parallel Repetition for GHZ-Supported Games via Spreadness. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{liu_et_al:LIPIcs.CCC.2026.6,
author = {Liu, Yang P. and Lovett, Shachar and Mittal, Kunal},
title = {{Improved Parallel Repetition for GHZ-Supported Games via Spreadness}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {6:1--6:21},
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.2026.6},
URN = {urn:nbn:de:0030-drops-270486},
doi = {10.4230/LIPIcs.CCC.2026.6},
annote = {Keywords: Parallel Repetition, GHZ Game, Algebraic Spreadness}
}
Document
The Log-Rank Conjecture: New Equivalent Formulations
Abstract
The log-rank conjecture is a longstanding open problem with multiple equivalent formulations in complexity theory and mathematics. In its linear-algebraic form, it asserts that the rank and partitioning number of a Boolean matrix are quasi-polynomially related.
We propose a relaxed but still equivalent version of the conjecture based on a new matrix parameter, signed rectangle rank: the minimum number of all-1 rectangles needed to express the Boolean matrix as a ± 1-sum. Signed rectangle rank lies between rank and partition number, and our main result shows that it is in fact equivalent to rank up to a logarithmic factor. Additionally, we extend the main result to tensors. This reframes the log-rank conjecture as: can every signed decomposition of a Boolean matrix be made positive with only quasi-polynomial blowup?
As an application, we prove an equivalence between the log-rank conjecture and a conjecture of Lovett and Singer–Sudan on cross-intersecting set systems.
Cite as
Lianna Hambardzumyan, Shachar Lovett, and Morgan Shirley. The Log-Rank Conjecture: New Equivalent Formulations. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 7:1-7:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hambardzumyan_et_al:LIPIcs.CCC.2026.7,
author = {Hambardzumyan, Lianna and Lovett, Shachar and Shirley, Morgan},
title = {{The Log-Rank Conjecture: New Equivalent Formulations}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {7:1--7:9},
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.2026.7},
URN = {urn:nbn:de:0030-drops-270495},
doi = {10.4230/LIPIcs.CCC.2026.7},
annote = {Keywords: cross-intersecting set systems, Log-rank conjecture, monochromatic rectangle, partition number}
}
Document
Efficient Adversaries
Abstract
The size of Frege proofs can be characterized in terms of prover-adversary games of Pudlák and Buss. We consider a generalization of prover-adversary games to many standard proof systems and show that some of the major proof complexity lower bounds such as the constant-depth Frege lower bound for the pigeonhole principle based on the method of k-evaluations, the Resolution lower bound for the weak pigeonhole principle based on the method of pseudo-width and Razborov’s Res(k) lower bound for Nisan-Wigderson generators based on expansion and a width lower bound (which is used to derive the Res(k)-hardness of formulas expressing circuit lower bounds) are constructive in the sense that they yield efficient algorithms computing winning strategies of adversaries in the generalized games. This is in contrast with our second result saying that if (a) such a constructive lower bound exists for Extended Frege system EF for formulas expressing succinct circuit lower bounds for SAT and (b) EF is strong enough to prove efficiently the correctness of anticheckers for SAT, then it is easy to separate the canonical pair of EF.
Cite as
Erfan Khaniki, Ján Pich, and Dmitry Sokolov. Efficient Adversaries. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 8:1-8:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{khaniki_et_al:LIPIcs.CCC.2026.8,
author = {Khaniki, Erfan and Pich, J\'{a}n and Sokolov, Dmitry},
title = {{Efficient Adversaries}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {8:1--8:32},
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.2026.8},
URN = {urn:nbn:de:0030-drops-270508},
doi = {10.4230/LIPIcs.CCC.2026.8},
annote = {Keywords: proof complexity, circuit complexity, lower bounds, barriers, truth-table formula, pigeonhole principle}
}
Document
Hardness of Computing Nondeterministic Kolmogorov Complexity
Abstract
Meta-complexity investigates the complexity of computational problems and tasks that are themselves about computations and their complexity. Understanding whether such problems can capture the hardness of NP is a central research direction. A longstanding open problem in this area is to establish the NP-hardness of MINKT (Ker-I Ko, 1991 [Ker{-}I Ko, 1991]), the problem of estimating time-bounded Kolmogorov complexity.
We contribute to this research direction by studying nK^t, a natural variant of Kolmogorov complexity that captures the complexity of representing a string using time-bounded nondeterministic computations [Buhrman et al., 2001]. Let MINnKT denote the task of estimating nK^t(x) of a given input string x. We prove that MINnKT ∈ BPP if and only if NP ⊆ BPP. This can be interpreted as a solution to Ko’s question in the setting of nondeterministic time-bounded Kolmogorov complexity.
Crucial to the proof of this result is the investigation of a new notion of probabilistic nondeterministic time-bounded Kolmogorov complexity called pnK^t. This measure can be seen as an extension of pK^t complexity [Halley Goldberg et al., 2022] obtained by replacing 𝖪^t with nK^t. We establish unconditionally that pnK^t has nearly all key properties of (time-unbounded) Kolmogorov complexity, such as language compression, conditional coding, and a form of symmetry of information. Finally, we show that the corresponding meta-computational problem MINpnKT also captures the hardness of NP, and that extending this result to the closely related problem Gap-MINpnKT would imply the exclusion of PH-Heuristica.
Cite as
Jinqiao Hu, Zhenjian Lu, and Igor C. Oliveira. Hardness of Computing Nondeterministic Kolmogorov Complexity. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 9:1-9:50, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hu_et_al:LIPIcs.CCC.2026.9,
author = {Hu, Jinqiao and Lu, Zhenjian and Oliveira, Igor C.},
title = {{Hardness of Computing Nondeterministic Kolmogorov Complexity}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {9:1--9:50},
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.2026.9},
URN = {urn:nbn:de:0030-drops-270516},
doi = {10.4230/LIPIcs.CCC.2026.9},
annote = {Keywords: meta-complexity, average-case complexity, Kolmogorov complexity}
}
Document
The Rate-Immediacy Barrier in Explicit Tree Code Constructions
Abstract
Since the introduction of tree codes by Schulman (STOC 1993), explicit construction of asymptotically good tree codes has remained a notorious challenge. A work by Cohen, Haeupler and Schulman (STOC 2018), as well as the state-of-the-art construction by Ben Yaacov, Cohen, and Yankovitz (STOC 2022) have achieved codes with rate Ω(1/log log n), exponentially improving upon the original rate Ω(1/log n) construction of Evans, Klugerman and Schulman from 1994. All of these constructions rely, at least in part, on increasingly sophisticated methods of combining (block) error-correcting codes.
In this work, we identify a fundamental barrier to constructing tree codes using known techniques. We introduce a key property which we call immediacy, that, while not required by the original definition of tree codes, is shared by all known constructions and inherently arises in recursive combinations of error-correcting codes. Our main technical contribution is the proof of a rate–immediacy trade-off, which, in particular, implies that any tree code with constant distance and non-trivial immediacy must necessarily have vanishing rate. By applying our rate-immediacy trade-off to existing constructions, we establish that their known rate analyses are essentially optimal given their actual error-correction properties. More broadly, our work highlights the need for fundamentally new ideas - beyond the recursive use of error-correcting codes - to achieve substantial progress in explicitly constructing asymptotically good tree codes.
Cite as
Gil Cohen, Leonard J. Schulman, and Piyush Srivastava. The Rate-Immediacy Barrier in Explicit Tree Code Constructions. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cohen_et_al:LIPIcs.CCC.2026.10,
author = {Cohen, Gil and Schulman, Leonard J. and Srivastava, Piyush},
title = {{The Rate-Immediacy Barrier in Explicit Tree Code Constructions}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {10:1--10:22},
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.2026.10},
URN = {urn:nbn:de:0030-drops-270522},
doi = {10.4230/LIPIcs.CCC.2026.10},
annote = {Keywords: Tree codes, Information Theory}
}
Document
Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?
Abstract
The complexity of bilinear maps (equivalently, of 3-mode tensors) has been studied extensively, most notably in the context of matrix multiplication. While circuit complexity and tensor rank coincide asymptotically for 3-mode tensors, this correspondence breaks down for d ≥ 4 modes. As a result, the complexity of d-mode tensors for larger fixed d remains poorly understood, despite its relevance, e.g., in fine-grained complexity. Our paper explores this intermediate regime.
First, we give a "graph-theoretic" proof of Strassen’s 2ω/3 bound on the asymptotic rank exponent of 3-mode tensors. Our proof directly generalizes to an upper bound of (d-1)ω/3 for d-mode tensors. Using refined techniques available only for d ≥ 4 modes, we improve this bound beyond the current state of the art for ω. We also obtain a bound of d/2+1 on the asymptotic exponent of circuit complexity of generic d-mode tensors and optimized bounds for d ∈ {4,5}.
To the best of our knowledge, asymptotic circuit complexity (rather than rank) of tensors has not been studied before. To obtain a robust theory, we first ask whether low complexity of T and U imply low complexity of their Kronecker product T ⊗ U. While this crucially holds for rank (and thus for circuit complexity in 3 modes), we show that assumptions from fine-grained complexity rule out such a submultiplicativity for the circuit complexity of tensors with many modes. In particular, assuming the Hyperclique Conjecture, this failure occurs already for d = 8 modes. Nevertheless, we can salvage a restricted notion of submultiplicativity.
From a technical perspective, our proofs heavily make use of the graph tensors T_H, as employed by Christandl and Zuiddam (Comput. Complexity 28 (2019) 27-56) and Christandl, Vrana and Zuiddam (Comput. Complexity 28 (2019) 57-111), whose modes correspond to the vertices of undirected graphs H. We make the simple but conceptually crucial observation that Kronecker products T_G ⊗ T_H are isomorphic to T_{G+H}, and that G and H may also be fractional graphs. By asymptotically converting generic tensors to specific graph tensors, we can use nontrivial results from algorithmic graph theory to study the rank and complexity of d-mode tensors for fixed d.
Cite as
Cornelius Brand, Radu Curticapean, Petteri Kaski, Baitian Li, Ian Orzel, Tim Seppelt, and Jiaheng Wang. Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{brand_et_al:LIPIcs.CCC.2026.11,
author = {Brand, Cornelius and Curticapean, Radu and Kaski, Petteri and Li, Baitian and Orzel, Ian and Seppelt, Tim and Wang, Jiaheng},
title = {{Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {11:1--11:23},
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.2026.11},
URN = {urn:nbn:de:0030-drops-270530},
doi = {10.4230/LIPIcs.CCC.2026.11},
annote = {Keywords: arithmetic circuits, tensor rank, bilinear complexity, graph tensors}
}
Document
Fine-Grained Complexity for Quantum Problems from Size-Preserving Circuit-To-Hamiltonian Constructions
Abstract
The local Hamiltonian (LH) problem is the canonical QMA-complete problem introduced by Kitaev. In this paper, we show its hardness in a very strong sense: we show that the 3-local Hamiltonian problem on n qubits cannot be solved classically in time O(2^{(1-ε)n}) for any ε > 0 under the Strong Exponential-Time Hypothesis (SETH), and cannot be solved quantumly in time O(2^{(1-ε)n/2}) for any ε > 0 under the Quantum Strong Exponential-Time Hypothesis (QSETH). These lower bounds give evidence that the currently known classical and quantum algorithms for LH cannot be significantly improved.
Furthermore, we are able to demonstrate fine-grained complexity lower bounds for approximating the quantum partition function (QPF) with an arbitrary constant relative error. Approximating QPF with relative error is known to be equivalent to approximately counting the dimension of the solution subspace of QMA problems. We show the SETH and QSETH hardness to estimate QPF with constant relative error. We then provide a quantum algorithm that runs in O(√{2ⁿ}) time for an arbitrary 1/poly(n) relative error, matching our lower bounds and improving the state-of-the-art algorithm by Bravyi, Chowdhury, Gosset, and Wocjan (Nature Physics 2022) in the low-temperature regime.
To prove our fine-grained lower bounds, we introduce the first size-preserving circuit-to-Hamiltonian construction that encodes the computation of a T-time quantum circuit acting on N qubits into a (d+1)-local Hamiltonian acting on N+O(T^{1/d}) qubits. This improves the standard construction based on the unary clock, which uses N+O(T) qubits.
Cite as
Nai-Hui Chia, Atsuya Hasegawa, François Le Gall, and Yu-Ching Shen. Fine-Grained Complexity for Quantum Problems from Size-Preserving Circuit-To-Hamiltonian Constructions. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 12:1-12:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chia_et_al:LIPIcs.CCC.2026.12,
author = {Chia, Nai-Hui and Hasegawa, Atsuya and Le Gall, Fran\c{c}ois and Shen, Yu-Ching},
title = {{Fine-Grained Complexity for Quantum Problems from Size-Preserving Circuit-To-Hamiltonian Constructions}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {12:1--12:35},
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.2026.12},
URN = {urn:nbn:de:0030-drops-270543},
doi = {10.4230/LIPIcs.CCC.2026.12},
annote = {Keywords: Fine-grain complexity, SETH, QSETH, Local Hamiltonian problem, Quantum partition problem}
}
Document
Resolution Width Lifts to Near-Quadratic-Depth Res(⊕) Size
Abstract
We show that for any unsatisfiable CNF formula φ that requires resolution refutation width at least w, and for any 1-stifling gadget g (for example, g = MAJ₃), (1) every resolution-over-parities (Res(⊕)) refutation of the lifted formula φ∘g of size at most S has depth at least Ω(w²/log S); (2) every Res(⊕) refutation of the lifted formula φ∘g has size Ω(w²).
The first result substantially extends and simplifies all previously known lifting theorems for bounded-depth Res(⊕). The lifting result of Itsykson and Knop [Dmitry Itsykson and Alexander Knop, 2026] requires gadgets of logarithmic size and applies only to refutations of depth at most O(nlog n), whereas our result applies to nearly quadratic depth. The liftings of Bhattacharya and Chattopadhyay [Sreejata Kishor Bhattacharya and Arkadev Chattopadhyay, 2025] and of Byramji and Impagliazzo [Farzan Byramji and Russell Impagliazzo, 2025] apply to nearly quadratic depth as well, but rely on a much stronger assumption of (Ω(n),Ω(n))-DT-hardness, which is far less standard than large resolution width.
Our proof combines the random-walk-with-restarts method of Alekseev and Itsykson [Yaroslav Alekseev and Dmitry Itsykson, 2025] with a new idea: the random walk is defined relative to the structure of the refutation graph, rather than by a distribution on inputs induced by the formula.
Using this technique, we substantially strengthen the supercritical size-depth tradeoff of Itsykson and Knop [Dmitry Itsykson and Alexander Knop, 2026], both by improving the depth lower bound and by reducing the size of the separating formulas to polynomial in the number of variables, with the latter resolving an open question posed in [Dmitry Itsykson and Alexander Knop, 2026]. In particular, we construct a family of polynomial-size formulas that admit polynomial-size resolution refutations, while any Res(⊕) refutation of depth o(n²/log⁴ n) necessarily has superpolynomial size.
Our second result yields a pure quadratic lower bound on the size of Res(⊕) refutations, improving upon the previously known near-quadratic lower bound of [Farzan Byramji and Russell Impagliazzo, 2025].
Cite as
Dmitry Itsykson, Vladimir Podolskii, and Alexander Shekhovtsov. Resolution Width Lifts to Near-Quadratic-Depth Res(⊕) Size. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 13:1-13:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{itsykson_et_al:LIPIcs.CCC.2026.13,
author = {Itsykson, Dmitry and Podolskii, Vladimir and Shekhovtsov, Alexander},
title = {{Resolution Width Lifts to Near-Quadratic-Depth Res(⊕) Size}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {13:1--13:27},
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.2026.13},
URN = {urn:nbn:de:0030-drops-270558},
doi = {10.4230/LIPIcs.CCC.2026.13},
annote = {Keywords: lifting theorems, resolution depth, resolution over parities, resolution width, supercritical tradeoff, random walk with restarts}
}
Document
A Weak Regularity Lemma for Polynomials
Abstract
A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming of having tower-type bounds or worse. In this paper we design a new, weaker regularity lemma with strong bounds. The new regularity lemma in particular provides means for quantitatively studying the curves contained in the image of a polynomial map, which is beyond the reach of standard methods.
The weak regularity lemma turns out to be powerful enough to yield results on arithmetic circuits and polynomial ranks that may be of independent interest:
- A general upper bound on the arithmetic circuit size of low-degree polynomials based solely on their image.
- An upper bound on the top fan-in of depth-4 arithmetic formulas under similar conditions.
- A quantitative bound for the Green-Tao notion of rank for polynomials, significantly improving on a result of Karam.
Cite as
Guy Moshkovitz and Dora Woodruff. A Weak Regularity Lemma for Polynomials. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 14:1-14:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{moshkovitz_et_al:LIPIcs.CCC.2026.14,
author = {Moshkovitz, Guy and Woodruff, Dora},
title = {{A Weak Regularity Lemma for Polynomials}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {14:1--14:24},
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.2026.14},
URN = {urn:nbn:de:0030-drops-270569},
doi = {10.4230/LIPIcs.CCC.2026.14},
annote = {Keywords: weak regularity lemma, finite-field polynomials, polynomial maps, structure-versus-randomness, arithmetic circuits}
}
Document
Derandomised Tensor Product Gap Amplification for Quantum Hamiltonians
Abstract
The quantum PCP conjecture asks whether it is QMA-hard to distinguish between high- and low-energy Hamiltonians even when the gap between "high" and "low" energy is large (constant). A natural proof strategy is gap amplification: start from the fact that high- and low-energy Hamiltonians are hard to distinguish if the gap is small (inverse polynomial) [Alexei Y. Kitaev et al., 2002] and amplify the Hamiltonians to increase the energy gap while preserving hardness. Such a gap amplification procedure is at the heart of Dinur’s proof of the classical PCP theorem [Dinur, 2007]. In this work, following Dinur’s model, we introduce a new quantum gap amplification procedure for Hamiltonians which uses random walks on expander graphs to derandomise (subsample the terms of) the tensor product amplification of a Hamiltonian. Curiously, our analysis relies on a new technique inspired by quantum de Finetti theorems, which have previously been used to rule out certain approaches to the quantum PCP conjecture [Fernando G. S. L. Brandão and Aram Wettroth Harrow, 2013].
Cite as
Thiago Bergamaschi, Tony Metger, Thomas Vidick, and Tina Zhang. Derandomised Tensor Product Gap Amplification for Quantum Hamiltonians. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bergamaschi_et_al:LIPIcs.CCC.2026.15,
author = {Bergamaschi, Thiago and Metger, Tony and Vidick, Thomas and Zhang, Tina},
title = {{Derandomised Tensor Product Gap Amplification for Quantum Hamiltonians}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {15:1--15:22},
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.2026.15},
URN = {urn:nbn:de:0030-drops-270572},
doi = {10.4230/LIPIcs.CCC.2026.15},
annote = {Keywords: quantum PCP conjecture, gap amplification, local Hamiltonians, tensor product amplification, expander random walks, quantum de Finetti theorems}
}
Document
Constant-Depth Circuits for Polynomial GCD over Any Characteristic
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
Rank Bounds and Polynomial-Time PIT for Σ^k Π Σ Π² Circuits
Abstract
A depth-4 algebraic circuit with top fan-in k and bottom fan-in 2 is a circuit Φ of the form Φ = ∑_{i = 1}^k ∏_{j = 1}^{m_i} Q_{ij}, where the polynomials Q_{ij} ∈ 𝕂[x₁, …, x_n] have degree at most 2. The class of all such circuits is denoted by Σ^k Π Σ Π². We say that the circuit Φ is an identity if it formally computes the zero polynomial. An important parameter of Σ^k Π Σ Π² circuits Φ is their (linear) rank, which is defined as the vector space dimension of the polynomials {Q_{ij}}_{i ∈ [k], j ∈ [m_i]}.
We prove that, when the base field 𝕂 is of characteristic zero, the rank of any (simple and minimal) Σ^k Π Σ Π² identity is upper bounded by a function which depends only on the top fan-in k. This result makes progress on [Beecken et al., 2013], being the first work to establish a bound on the rank of such identities that depends only on the top fan-in. Moreover, when combined with [Beecken et al., 2013], our main result yields the first deterministic, polynomial time PIT algorithm for Σ^k Π Σ Π² circuits.
One of the key components of our proof of the rank bounds is the derivation of an approximate Hansen-type result, which is interesting in its own right. This result can be seen as an algebraic and higher-dimensional analogue of the approximate Sylvester-Gallai result of [Ai et al., 2014], and a distinct approximate fractional Sylvester-Gallai result than the one from [Garg et al., 2023]. Additionally, we prove a robust version of it, in the spirit of the generalization of Hansen’s theorem by [Boaz Barak et al., 2013].
This paper is an extended abstract of the full version of the paper, which can be found at [Garg et al., 2026].
Cite as
Abhibhav Garg, Rafael Oliveira, Akash Kumar Sengupta, Nir Shalmon, and Amir Shpilka. Rank Bounds and Polynomial-Time PIT for Σ^k Π Σ Π² Circuits. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{garg_et_al:LIPIcs.CCC.2026.17,
author = {Garg, Abhibhav and Oliveira, Rafael and Sengupta, Akash Kumar and Shalmon, Nir and Shpilka, Amir},
title = {{Rank Bounds and Polynomial-Time PIT for \Sigma^k \Pi \Sigma \Pi² Circuits}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {17:1--17:18},
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.2026.17},
URN = {urn:nbn:de:0030-drops-270599},
doi = {10.4230/LIPIcs.CCC.2026.17},
annote = {Keywords: Sylvester-Gallai Theorems, Polynomial Identity Testing, Strong Algebras}
}
Document
Multiplicative Pseudorandom Generators for Nondeterministic Circuits
Abstract
The hardness vs. randomness paradigm aims to construct pseudorandom generators (PRGs) based on complexity theoretic hardness assumptions. A seminal result in this area is a PRG construction by [N. Nisan and A. Wigderson, 1994; R. Impagliazzo and A. Wigderson, 1997]. A sequence of works [A. Klivans and D. van Melkebeek, 2002; R. Shaltiel and C. Umans, 2005; C. Umans, 2003; R. Shaltiel and C. Umans, 2006] generalized the result of [N. Nisan and A. Wigderson, 1994; R. Impagliazzo and A. Wigderson, 1997] to nondeterministic circuits, and showed that if E = DTIME(2^{O(n)}) requires nondeterministic circuits of size 2^{Ω(n)}, then for every sufficiently large s, and every ε ≥ 1/s, there is an ε-PRG G:{0,1}^{r = O(log s + log 1/(ε))} → {0,1}^s that runs in time poly(s), and fools size s nondeterministic circuits. In particular, for every size s nondeterministic circuit C,
Pr[C(G(U_r)) = 1] ≤ Pr[C(U_s) = 1] + ε.
Applebaum et al. [B. Applebaum et al., 2015] showed that "black-box techniques" cannot achieve such results for ε = s^{-ω(1)}. In order to circumvent this problem, Artemenko et al. [S. Artemenko et al., 2016] suggested a "multiplicative" version of PRGs, which requires that:
Pr[C(G(U_r)) = 1] ≤ 2 ⋅ Pr[C(U_s) = 1] + ε.
This still gives that Pr[C(G(U_r)) = 1] is very small, if Pr[C(U_s) = 1] is very small, and is therefore suitable for applications that only require this consequence. [S. Artemenko et al., 2016] constructed such multiplicative PRGs for ε = s^{-ω(1)} (based on very strong hardness assumptions).
In this paper, we give an optimal construction of multiplicative PRGs for nondeterministic circuits. More specifically, under the same hardness assumption used for (standard) PRGs for nondeterministic circuits, we show that for every ε ≥ 1/(2^{s)}, there is a multiplicative PRG G:{0,1}^{r = O(log s + log 1/(ε))} → {0,1}^s that runs in time poly(s) and fools size s nondeterministic circuits.
This gives the optimal seed length under a hardness assumption that is necessary, and provides improvements in several applications of multiplicative PRGs. Our result improves upon the previous multiplicative PRG construction of [S. Artemenko et al., 2016], which uses a stronger hardness assumption against Σ₃-circuits, and where the seed length is the suboptimal r = O(log s) + O(log 1/(ε))². Our result also improves upon the recent multiplicative PRG of Shaltiel [R. Shaltiel, 2025] that only achieves very small stretch (the output length in [R. Shaltiel, 2025] is less than twice the seed length).
Our PRG construction borrows ideas from the recent "low stretch" PRG of Shaltiel [R. Shaltiel, 2025], and the (standard) PRG construction of Shaltiel and Umans [R. Shaltiel and C. Umans, 2005]. Loosely speaking, we aim to get the "multiplicativity" of the former, and the "large stretch" of the latter. While both approaches generalize the list-decoding results of Sudan, Trevisan and Vadhan [M. Sudan et al., 2001], the two results are tailored to two very different parameter regimes, and we introduce several new ideas to make the two approaches co-exist.
Cite as
Alon Dermer and Ronen Shaltiel. Multiplicative Pseudorandom Generators for Nondeterministic Circuits. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dermer_et_al:LIPIcs.CCC.2026.18,
author = {Dermer, Alon and Shaltiel, Ronen},
title = {{Multiplicative Pseudorandom Generators for Nondeterministic Circuits}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {18:1--18:20},
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.2026.18},
URN = {urn:nbn:de:0030-drops-270602},
doi = {10.4230/LIPIcs.CCC.2026.18},
annote = {Keywords: Hardness vsRandomness, Pseudorandomness, Nondeterministic Circuits}
}
Document
When Hilbert Approximates: A Strong Nullstellensatz for Approximate Polynomial Satisfiability
Abstract
Guo, Saxena, and Sinhababu (TOC'18, CCC'18) defined a natural, approximative analog of the polynomial system satisfiability problem, which they called approximate polynomial satisfiability (APS). They proved algebraic and geometric properties of it and showed an NP-hardness lower bound and a PSPACE upper bound for it. They further established how the problem naturally occurs in border complexity and Geometric complexity theory (GCT) and used the problem to construct hitting sets for ̅{VP} in PSPACE, hence greatly mitigating the GCT chasm.
The starting point of this work is the observation that Guo, Saxena, and Sinhababu’s criterion for non-existence of approximative solution can be interpreted as an analog of Weak Hilbert’s Nullstellensatz in the approximative setting. We extend their work by proving an analog of Strong Hilbert’s Nullstellensatz in the approximative setting. Concretely, we give an algebraic criterion for containment between approximative solution sets defined by systems of polynomials. In fact, this characterization turns out to be equivalent to membership in the integral closure over a maximal ideal of a local subring of ℂ(x₁,…, x_n) determined by the given polynomials. In addition, we use our proof to provide a PSPACE algorithm for testing this containment, exponentially better than the EXPSPACE bounds for polynomial subalgebra membership testing and the polynomial integral closure membership testing, hence matching the complexity bound of Guo, Saxena, and Sinhababu’s Weak Approximative Nullstellensatz.
Cite as
Sanyam Agarwal, Sagnik Dutta, Anurag Pandey, and Himanshu Shukla. When Hilbert Approximates: A Strong Nullstellensatz for Approximate Polynomial Satisfiability. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{agarwal_et_al:LIPIcs.CCC.2026.19,
author = {Agarwal, Sanyam and Dutta, Sagnik and Pandey, Anurag and Shukla, Himanshu},
title = {{When Hilbert Approximates: A Strong Nullstellensatz for Approximate Polynomial Satisfiability}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {19:1--19:18},
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.2026.19},
URN = {urn:nbn:de:0030-drops-270618},
doi = {10.4230/LIPIcs.CCC.2026.19},
annote = {Keywords: Approximate Polynomial Satisfiability, Algebraic Geometry, PSPACE}
}
Document
On Factorization of Sparse Polynomials of Bounded Individual Degree
Abstract
We study sparse polynomials with bounded individual degree and the class of their factors. In particular we obtain the following algorithmic and structural results:
1) A deterministic polynomial-time algorithm for finding all the sparse divisors of a sparse polynomial with bounded individual degree. As part of this, we establish the first upper bound on the number of non-monomial irreducible factors of such polynomials.
2) A poly(n,s^{dlog 𝓁})-time algorithm for recovering 𝓁 irreducible s-sparse polynomials of bounded individual degree d from blackbox access to their product (which is not necessarily sparse). This partially resolves a question posed in [Pranjal Dutta et al., 2024]. In particular, when 𝓁 = O(1), the algorithm runs in polynomial time.
3) Deterministic algorithms for factoring a product of s-sparse polynomials of bounded individual degree d from blackbox access. Over fields of characteristic zero or sufficiently large, the algorithm runs in poly(n,s^{d³log n})-time; over arbitrary fields it runs in poly(n,{(d²)!},s^{d⁵log n})-time. This improves upon the algorithm of [Bhargava et al., 2020], which runs in poly(n,s^{d⁷log n})-time and applies only to a single sparse polynomial of bounded individual degree. In the case where the input is a single sparse polynomial, we give an algorithm that runs in poly(n,s^{d²log n})-time.
4) Given blackbox access to a product of (not necessarily sparse or irreducible) factors of sparse polynomials of bounded individual degree, we give a deterministic polynomial-time algorithm for finding all irreducible sparse multiquadratic factors of it (along with their multiplicities). This generalizes the algorithms of [Volkovich, 2015] and [Volkovich, 2017]. We also show how to decide whether such a product is a complete power (in case it is defined over a field of zero or large enough characteristic), extending the algorithm of [Bisht and Volkovich, 2025].
Our algorithms most naturally apply over fields of zero or sufficiently large characteristic. To handle arbitrary fields, we introduce the notion of primitive divisors for a class of polynomials, which may be of independent interest. This notion enables us to adapt ideas of [Bisht and Volkovich, 2025] and remove characteristic assumptions from most of our algorithms.
Cite as
Aminadav Chuyoon and Amir Shpilka. On Factorization of Sparse Polynomials of Bounded Individual Degree. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chuyoon_et_al:LIPIcs.CCC.2026.20,
author = {Chuyoon, Aminadav and Shpilka, Amir},
title = {{On Factorization of Sparse Polynomials of Bounded Individual Degree}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {20:1--20:20},
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.2026.20},
URN = {urn:nbn:de:0030-drops-270627},
doi = {10.4230/LIPIcs.CCC.2026.20},
annote = {Keywords: algebraic complexity theory, sparse polynomials, factorization, reconstruction}
}
Document
Wide Replacement Products Meet Gray Codes: Toward Optimal Small-Bias Sets
Abstract
Optimal small-bias sets sit at the crossroads of coding theory and pseudorandomness. Reaching optimal parameters would, in particular, meet the long-standing goal of matching the Gilbert-Varshamov bound for binary codes in the high-distance regime. In a breakthrough, Ta-Shma [Ta-Shma, 2017] constructed near-optimal small-bias sets via the Rozenman-Wigderson expander-walk framework, using the wide-replacement product to maintain s "secure" registers and to route the walk through them. Within this framework, two barriers remain en route to optimal small-bias sets: (i) the cost of maintaining registers and (ii) limitations inherited from spectral-gap bounds for expanders.
We overcome the first - arguably the more critical - barrier. Our key technical insight is that registers can be reused even after they are exposed. Using a Gray-code-style reuse schedule, we recycle the same s registers exponentially many times in s, thereby reducing the register-maintenance cost exponentially. This yields the first improvement over Ta-Shma’s construction in nearly a decade - quantitatively modest but an important first step toward a truly optimal construction. The remaining barrier is fairly standard in isolation; the challenge is to overcome it in concert with our register-reuse framework.
Cite as
Gil Cohen and Itay Cohen. Wide Replacement Products Meet Gray Codes: Toward Optimal Small-Bias Sets. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 21:1-21:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cohen_et_al:LIPIcs.CCC.2026.21,
author = {Cohen, Gil and Cohen, Itay},
title = {{Wide Replacement Products Meet Gray Codes: Toward Optimal Small-Bias Sets}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {21:1--21:27},
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.2026.21},
URN = {urn:nbn:de:0030-drops-270631},
doi = {10.4230/LIPIcs.CCC.2026.21},
annote = {Keywords: small-bias sets, bias amplification, expansion-amplification, gray codes}
}
Document
Multilinear Algebraic Branching Programs and the Min-Partition Rank Method
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
Quantum-Classical Equivalence for And-Functions
Abstract
A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for And-functions of the form
F(x,y) = f(x₁ ∧ y₁, …, x_n ∧ y_n),
when the outer function f is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all And-functions has remained open and has been posed by several authors.
In this work, we settle this problem in a strong way. We show that for every Boolean function f, the bounded-error quantum and classical deterministic communication complexities of the function f∘And₂ are polynomially related, up to polylogarithmic factors in n. We prove this by showing that both are characterized - up to polynomial loss - by the logarithm of the De Morgan sparsity of f.
Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett [Arkadev Chattopadhyay et al., 2026] on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general And-functions.
Cite as
Sreejata Kishor Bhattacharya, Farzan Byramji, Arkadev Chattopadhyay, Yogesh Dahiya, and Shachar Lovett. Quantum-Classical Equivalence for And-Functions. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 23:1-23:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhattacharya_et_al:LIPIcs.CCC.2026.23,
author = {Bhattacharya, Sreejata Kishor and Byramji, Farzan and Chattopadhyay, Arkadev and Dahiya, Yogesh and Lovett, Shachar},
title = {{Quantum-Classical Equivalence for And-Functions}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {23:1--23:24},
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.2026.23},
URN = {urn:nbn:de:0030-drops-270656},
doi = {10.4230/LIPIcs.CCC.2026.23},
annote = {Keywords: Communication complexity, quantum communication complexity, De Morgan sparsity, approximate gamma two norm, And-functions}
}
Document
All Polynomial Generators Preserve Distance with Mutual Correlated Agreement
Abstract
A generator is a function that maps a random seed to a list of coefficients. We study generators that preserve distance to a linear code: the linear combination of any list of vectors using coefficients sampled by the generator has distance to the code no smaller than that of the original vectors, except for a small error. Distance preservation plays a central role in modern probabilistic proofs, and has been formalized in several ways. We study mutual correlated agreement, the strongest known form of distance preservation.
We initiate a systematic study of mutual correlated agreement, aiming to characterize the class of generators with this property. Towards this, we study polynomial generators, a rich class that includes all examples of generators considered in the distance preservation literature. Our main result is that all polynomial generators guarantee mutual correlated agreement for every linear code. This improves on prior work both in generality (the class of generators covered) and in parameters (the error bounds).
We additionally provide new results for the case where the linear code is a Reed-Solomon code, which is of particular interest in applications. We prove that all polynomial generators satisfy mutual correlated agreement for Reed-Solomon codes up to the Johnson bound. In particular, we improve upon the state-of-the-art by Ben-Sasson, Carmon, Ishai, Kopparty, and Saraf (FOCS 2020) and answer a question posed by Arnon, Chiesa, Fenzi, and Yogev (Eurocrypt 2025).
Along the way we develop a flexible and general toolbox for mutual correlated agreement, and are the first to establish distance preservation for generators that lie beyond polynomial generators.
Cite as
Sarah Bordage, Alessandro Chiesa, Ziyi Guan, and Ignacio Manzur. All Polynomial Generators Preserve Distance with Mutual Correlated Agreement. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bordage_et_al:LIPIcs.CCC.2026.24,
author = {Bordage, Sarah and Chiesa, Alessandro and Guan, Ziyi and Manzur, Ignacio},
title = {{All Polynomial Generators Preserve Distance with Mutual Correlated Agreement}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {24:1--24:18},
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.2026.24},
URN = {urn:nbn:de:0030-drops-270666},
doi = {10.4230/LIPIcs.CCC.2026.24},
annote = {Keywords: proximity testing, distance preservation, mutual correlated agreement}
}
Document
Polynomial Identity Testing for Read-4 Arithmetic Formulas
Abstract
We present the first algorithms for polynomial identity testing (PIT) of read-4 arithmetic formulas in the non-multilinear setting. Specifically, we give a polynomial-time PIT algorithm in the whitebox model and a quasi-polynomial-time algorithm in the blackbox model. Since our techniques are based on proving hardness of representation results, we extend our algorithms to orbits of read-4 formulas under the action of the affine linear group. Prior to our work, no subexponential white- or blackbox algorithms were known for this class of formulas. All our results hold over any field 𝔽 with char(𝔽) = 0 or char(𝔽) ≥ 5.
Prior work addressed only restricted cases. Anderson, van Melkebeek, and Volkovich (Computational Complexity, 2015) studied multilinear read-k formulas, giving a polynomial-time whitebox PIT algorithm and a quasi-polynomial-time blackbox algorithm. Without the multilinearity restriction, Mahajan, Rao, and Sreenivasaiah (TCS, 2014) gave polynomial-time whitebox algorithms for read-2 and read-3 formulas, Prakriya (Doctoral Thesis, 2019) obtained quasi-polynomial-time blackbox PIT algorithm for both read-2 and read-3 formulas. Independently, Shamir (Master’s Thesis, 2022) obtained a quasi-polynomial-time blackbox PIT algorithm for read-2 formulas. For bounded-depth read-k formulas, Agrawal, Saha, Saptharishi, and Saxena (SICOMP, 2016) obtained a polynomial-time blackbox algorithm in the non-multilinear case. The running time of their algorithm is n^{k^{2^Δ}} for read-k, depth-Δ formulas, and hence it is applicable only to constant depth.
Partial derivatives are a central tool in the study of deterministic PIT for bounded-read formulas. However, for non-multilinear RkF, differentiation may increase the number of reads. To address this, we develop new structural results that ensure "nice behavior" of derivatives. Specifically, we introduce a new Fragmentation Lemma that reduces the PIT problem for general RkFs to simpler models via differentiation. In addition, we define the notion of dominating degree patterns and show that, in certain cases, taking partial derivatives with respect to these patterns preserves the read count.
Cite as
Nimrod Kaplan and Amir Shpilka. Polynomial Identity Testing for Read-4 Arithmetic Formulas. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 25:1-25:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kaplan_et_al:LIPIcs.CCC.2026.25,
author = {Kaplan, Nimrod and Shpilka, Amir},
title = {{Polynomial Identity Testing for Read-4 Arithmetic Formulas}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {25:1--25:18},
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.2026.25},
URN = {urn:nbn:de:0030-drops-270678},
doi = {10.4230/LIPIcs.CCC.2026.25},
annote = {Keywords: algebraic complexity theory, polynomial identity testing, PIT, bounded read formulas}
}
Document
A Simple Sub-Polynomial Degree Coboundary Expander
Abstract
High dimensional expanders simultaneously satisfying spectral and combinatorial (coboundary) expansion have recently played a major role in breakthroughs in PCP and coding theory, but the only known construction of such complexes is extremely involved, requiring deep algebraic number theory. In this work, we give an extremely simple combinatorial construction of a sub-polynomial degree complex based on projections of the flags complex (subspace chains) that is (i) a local spectral expander, (ii) a coboundary expander, and (iii) a swap coboundary expander. As a corollary, we also give the first near-linear size combinatorial hypergraphs with good agreement tests in the `1%' regime, and a simple PCP construction with near-linear size.
Cite as
Max Hopkins and Arka Ray. A Simple Sub-Polynomial Degree Coboundary Expander. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 26:1-26:41, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hopkins_et_al:LIPIcs.CCC.2026.26,
author = {Hopkins, Max and Ray, Arka},
title = {{A Simple Sub-Polynomial Degree Coboundary Expander}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {26:1--26:41},
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.2026.26},
URN = {urn:nbn:de:0030-drops-270685},
doi = {10.4230/LIPIcs.CCC.2026.26},
annote = {Keywords: high dimensional expansders, agreement testing, probabilistically checkable proofs}
}
Document
Optimal Depth-Three Circuits for Inner Product
Abstract
We show that Inner Product in 2n variables, IP_n(x, y) = x₁y₁ ⊕ … ⊕ x_ny_n, can be computed by depth-3 bottom fan-in 2 circuits of size poly(n)⋅ (9/5)ⁿ, matching the lower bound of Göös, Guan, and Mosnoi (Inform. Comput.'24). Our construction is obtained via the following steps.
1) We provide a general template for constructing optimal depth-3 circuits with bottom fan-in k for an arbitrary function f. We do this in two steps. First, we partition f^{-1}(1) into orbits of its automorphism group. Second, for each orbit, we construct one k-CNF that (a) accepts the largest number of inputs from that orbit and (b) rejects all inputs rejected by f.
2) We instantiate the template for IP_n and k = 2. Guided by the intuition (which we call modularity principle) that optimal 2-CNFs can be constructed by taking the conjunction of variable-disjoint copies of smaller 2-CNFs, we use computer search to identify a small set of building block 2-CNFs over at most 4 variables.
3) We again use computer search to discover appropriate combinations (disjoint conjunctions) of building blocks to arrive at optimal 2-CNFs and analyze them using techniques from analytic combinatorics. We believe that the approach outlined in this paper can be applied to a wide range of functions to determine their depth-3 complexity.
Cite as
Mohit Gurumukhani, Daniel Kleber, Ramamohan Paturi, Christopher Rosin, and Navid Talebanfard. Optimal Depth-Three Circuits for Inner Product. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 27:1-27:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gurumukhani_et_al:LIPIcs.CCC.2026.27,
author = {Gurumukhani, Mohit and Kleber, Daniel and Paturi, Ramamohan and Rosin, Christopher and Talebanfard, Navid},
title = {{Optimal Depth-Three Circuits for Inner Product}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {27:1--27:25},
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.2026.27},
URN = {urn:nbn:de:0030-drops-270693},
doi = {10.4230/LIPIcs.CCC.2026.27},
annote = {Keywords: Depth 3 circuits, Circuit lower bounds, Inner product, Analytic combinatorics}
}
Document
Plethysm is in #BQP
Abstract
Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #𝖯. Whether this holds more generally is considered an important open problem in mathematics and computer science, with relevance for geometric complexity theory and quantum information. Recent work has investigated the quantum complexity of particular multiplicities, such as the Kronecker coefficients and certain special cases of the plethysm coefficients.
Here, we show that a broad class of representation-theoretic multiplicities is in #BQP. This includes the result that plethysm coefficients are in #BQP, which was only known in certain cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, thus unifying, simplifying, and extending prior work. We obtain our result by multiple applications of the Schur transform; recent work has improved its dependence on the local dimension, which is crucial for our work. We further describe a general approach for showing that representation-theoretic multiplicities are in #BQP that captures the approaches of our and previous work. We complement the above by showing that the same multiplicities are also naturally in GapP and obtain polynomial-time classical algorithms when certain parameters are fixed.
Cite as
Matthias Christandl, Aram W. Harrow, Greta Panova, Pietro M. Posta, and Michael Walter. Plethysm is in #BQP. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 28:1-28:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{christandl_et_al:LIPIcs.CCC.2026.28,
author = {Christandl, Matthias and Harrow, Aram W. and Panova, Greta and Posta, Pietro M. and Walter, Michael},
title = {{Plethysm is in #BQP}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {28:1--28:11},
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.2026.28},
URN = {urn:nbn:de:0030-drops-270706},
doi = {10.4230/LIPIcs.CCC.2026.28},
annote = {Keywords: Quantum witness counting, #BQP, algebraic combinatorics, representation theory, representation-theoretic multiplicities}
}
Document
Frontier Space-Time Algorithms Using Only Full Memory
Abstract
We develop catalytic algorithms for fundamental problems in algorithm design that run in polynomial time, use only 𝒪(log(n)) workspace, and use sublinear catalytic space matching the best-known space bounds of non-catalytic algorithms running in polynomial time.
First, we design a polynomial time algorithm for directed s-t connectivity using n / 2^{Θ(√{log n})} catalytic space, which matches the state-of-the-art time-space bounds in the non-catalytic setting [Barnes et al., 1998], and improves the catalytic space usage of the best known algorithm [James Cook and Edward Pyne, 2026]. Furthermore, using only 𝒪(log(n)) random bits we get a randomized algorithm whose running time nearly matches the fastest time bounds known for space-unrestricted algorithms.
Second, we design polynomial time algorithms for the problems of computing Edit Distance, Longest Common Subsequence, and the Discrete Fréchet Distance, again using n / 2^{Θ(√{log n})} catalytic space. This again matches non-catalytic time-space frontier for Edit Distance and Least Common Subsequence [Kiyomi et al., 2021].
Cite as
Petr Chmel, Aditi Dudeja, Michal Koucký, Ian Mertz, and Ninad Rajgopal. Frontier Space-Time Algorithms Using Only Full Memory. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 29:1-29:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chmel_et_al:LIPIcs.CCC.2026.29,
author = {Chmel, Petr and Dudeja, Aditi and Kouck\'{y}, Michal and Mertz, Ian and Rajgopal, Ninad},
title = {{Frontier Space-Time Algorithms Using Only Full Memory}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {29:1--29:23},
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.2026.29},
URN = {urn:nbn:de:0030-drops-270710},
doi = {10.4230/LIPIcs.CCC.2026.29},
annote = {Keywords: Catalytic computation, Connectivity, Edit distance, Discrete Fr\'{e}chet distance}
}
Document
Bounds for Hardness Condensation in the Query Model
Abstract
For any Boolean function f:{0,1}ⁿ → {0,1} with a complexity measure having value k ≪ n, is it possible to restrict the function f to Θ(k) variables while keeping the complexity preserved at Θ(k)? Instantiation of this question for the measure of circuit complexity of the Boolean function was shown to be related to circuit lower bounds (Buresh-Oppenheim and Santhanam, 2006). Variants of the above question were also shown to have connections to the log-rank conjecture in communication complexity (Hrubeš, 2024) and lower bounds in proof complexity (Razborov, 2016). In the context of communication and query complexity, this question was recently studied by Göös, Newman, Riazanov and Sokolov (2024). They showed, among other results, that query complexity cannot be condensed losslessly.
In this work, we show that there exists a Boolean function f such that any restriction of f to O(ℳ(f)) variables has ℳ(⋅)-complexity at most Õ(ℳ(f)^{2/3}), where ℳ is one of block sensitivity (bs), fractional block sensitivity (fbs), certificate complexity (𝖢), deterministic query complexity (𝖣), zero-error randomized query complexity (𝖱₀), and AND (and OR)-decision tree query complexity. This improves upon the results of Göös, Newman, Riazanov, and Sokolov (2024) for 𝖣 and 𝖱₀, and in particular answers their open question about the condensation of block sensitivity.
We complement the negative results on lossless condensation with positive results about lossy condensation. In particular, we show that for every Boolean function f there exists a restriction of f to O(ℳ(f)) variables such that its ℳ(⋅)-complexity is at least Ω(ℳ(f)^{1/2}), where ℳ ∈ {bs,fbs,𝖢,UC_{min},UC₁,UC,𝖣,deg̃,λ}. In addition, we show lossy condensation for randomized and quantum query complexity with a slightly smaller exponent.
Cite as
Chandrima Kayal, Rajat Mittal, Sai Soumya Nalli, Manaswi Paraashar, Karthikeya Polisetty, Jayalal Sarma, and Nitin Saurabh. Bounds for Hardness Condensation in the Query Model. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kayal_et_al:LIPIcs.CCC.2026.30,
author = {Kayal, Chandrima and Mittal, Rajat and Nalli, Sai Soumya and Paraashar, Manaswi and Polisetty, Karthikeya and Sarma, Jayalal and Saurabh, Nitin},
title = {{Bounds for Hardness Condensation in the Query Model}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {30:1--30:20},
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.2026.30},
URN = {urn:nbn:de:0030-drops-270722},
doi = {10.4230/LIPIcs.CCC.2026.30},
annote = {Keywords: Query Complexity, Decision Trees, Hardness Condensation}
}
Document
Optimal Testing of Reed-Muller Codes with an Online Adversary
Abstract
Motivated by applications to property testing in the online-erasure model of Kalemaj, Raskhodnikova, and Varma (ITCS 2022 and Theory of Computing 2023), we define and analyze semi-sample-based testers for Reed-Muller codes. The task in Reed-Muller testing is to determine whether an input function f: 𝔽ⁿ → 𝔽 belongs to the Reed-Muller code or is far from it, using as few point queries to f as possible. Reed-Muller testing is a well-studied task with its roots in both the Property Testing and Probabilistically Checkable Proofs literature. The online-erasure model introduces a twist: after each query made, an adversary may erase up to t points of the input function, potentially thwarting any test in which the queries follow a predictable pattern.
Semi-sample-based testers are a hybrid between sample-based testers - which can only make uniformly random queries to the input function - and standard testers, which can choose their queries freely. They are designed with the online-erasure model in mind and operate by first choosing some subset S of the domain and then making their queries uniformly at random inside of S. We describe semi-sample-based testers for the Reed-Muller code and give an optimal analysis of their soundness.
Consequently, we show that semi-sample-based testers are indeed effective in the presence of online erasures, and thereby achieve optimal query complexity for testing the Reed-Muller code in the online-erasure model. This result improves upon prior work of Minzer and Zheng (SODA 2024). As an added bonus, we show that semi-sample-based testers also exist for the lifted affine-invariant codes of Guo, Kopparty, and Sudan (ITCS 2013), thereby providing the first known testers for these codes in the online-erasure model.
Cite as
Esty Kelman, Uri Meir, and Kai Zhe Zheng. Optimal Testing of Reed-Muller Codes with an Online Adversary. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 31:1-31:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kelman_et_al:LIPIcs.CCC.2026.31,
author = {Kelman, Esty and Meir, Uri and Zheng, Kai Zhe},
title = {{Optimal Testing of Reed-Muller Codes with an Online Adversary}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {31:1--31:26},
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.2026.31},
URN = {urn:nbn:de:0030-drops-270732},
doi = {10.4230/LIPIcs.CCC.2026.31},
annote = {Keywords: Property testing, Low degree testing}
}
Document
Fixed-Parameter Degree Bounds and Complexity of the Orbit Closure Intersection Problem for Tensors
Abstract
The orbit closure intersection problem for a reductive group action is a geometric relaxation of the orbit equality problem. These problems capture a range of isomorphism, degeneration and identity-testing problems in computational complexity. We study this problem for the tensor action of G = SL_n(𝕂) x SL_n(𝕂) x SL_m(𝕂) on 𝒱 = 𝕂ⁿ⊗𝕂ⁿ⊗𝕂^m (equivalently, on m-tuples of n× n-matrices) where the base field is algebraically closed with characteristic zero.
We focus on the fixed-parameter regime where m is constant and n is allowed to grow. The case of m = 3 is already interesting in the context of tensor rank and matrix multiplication. Prior to our work, only a special case of this problem, namely when one of the input tensors is the zero-tensor (this corresponds to the null cone problem for the tensor action), was known to be solvable in polynomial time (Bürgisser-Franks-Garg-Oliveira-Walter-Wigderson, FOCS'19).
Our main result is to show that the orbit closure intersection problem for the above action can be solved in randomized polynomial time. This is achieved by the following new ingredients:
1) We prove an explicit fixed-parameter bound on the degrees needed to generate the invariant ring for the tensor action: for every fixed m, these bounds are polynomial in n.
2) We prove that there is a succinct encoding of the invariants: we construct a uniform, polynomial-sized arithmetic circuit generating all invariants up to the required degree.
Cite as
M. Levent Doğan, John Maar, Rafael Oliveira, and Youming Qiao. Fixed-Parameter Degree Bounds and Complexity of the Orbit Closure Intersection Problem for Tensors. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 32:1-32:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dogan_et_al:LIPIcs.CCC.2026.32,
author = {Do\u{g}an, M. Levent and Maar, John and Oliveira, Rafael and Qiao, Youming},
title = {{Fixed-Parameter Degree Bounds and Complexity of the Orbit Closure Intersection Problem for Tensors}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {32:1--32:21},
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.2026.32},
URN = {urn:nbn:de:0030-drops-270741},
doi = {10.4230/LIPIcs.CCC.2026.32},
annote = {Keywords: computational invariant theory, geometric complexity theory, orbit closure intersection problem}
}
Document
Multilinear Formula Lower Bounds for Sparse Determinants
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
Quantum Advantage in Tolerant Junta Testing
Abstract
We establish the first super-polynomial quantum advantage for the tolerant junta testing problem in the adaptive setting. Specifically, we show that within a certain parameter regime, tolerant k-junta testing with high precision can be solved using poly(k) quantum queries, whereas any classical algorithm requires at least k^{Ω(log k)} queries.
The problem of tolerant k-junta testing is as follows: given parameters (k, ε₁, ε₂), with 0 ≤ ε₁ < ε₂ ≤ 1/2, and black-box access to a Boolean function f (defined on n variables), distinguish whether f is ε₁-close to some k-junta or ε₂-far from every k-junta.
We show the quantum advantage for a range of parameters close to 1/2, for example, ε₁ = 1/2-1/k and ε₂ = 1/2-1/(2k²). (As such, the problem is more naturally captured using the notion of correlation with closest k-junta.) The (non-adaptive) quantum tester we use was given by a recent work of Bao, Liu, Yao, Ye, and Zhang (SOSA 2026). We slightly adapt their analysis to show that it holds in the above parameter regime. On the other hand, our classical lower bound requires substantial new ideas. Inspired by the lower bound techniques of Chen and Patel (FOCS 2023), we introduce a new hard distribution of "yes" instances (i.e., instances with distance at most ε₁ to k-juntas) that is based on planting an "approximate-junta" as follows: we randomly pick k out of n coordinates, and for each fixing of the k coordinates, the 2^{n-k} values in the restricted subcube are drawn randomly except for an error-correcting code on which we place the same random bit. We show that this distribution is much closer to k-juntas than the uniform distribution, but on the other hand, they are indistinguishable with respect to any classical algorithm making k^{o(log k)} queries.
Cite as
Avishay Tal and Weiqiang Yuan. Quantum Advantage in Tolerant Junta Testing. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{tal_et_al:LIPIcs.CCC.2026.34,
author = {Tal, Avishay and Yuan, Weiqiang},
title = {{Quantum Advantage in Tolerant Junta Testing}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {34:1--34:17},
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.2026.34},
URN = {urn:nbn:de:0030-drops-270762},
doi = {10.4230/LIPIcs.CCC.2026.34},
annote = {Keywords: Property testing, Quantum query complexity, Error-correcting code, Boolean function analysis}
}
Document
Trace Hermitian Codes Have Vanishing Bias
Abstract
In this work we give the first proof that Trace Hermitian codes have vanishing bias. This brings to the front the question of understanding the distance of Trace AG codes, and the fascinating possibility that in some variation they might give asymptotically good codes.
Cite as
Swastik Kopparty, Amnon Ta-Shma, and Kedem Yakirevitch. Trace Hermitian Codes Have Vanishing Bias. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 35:1-35:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kopparty_et_al:LIPIcs.CCC.2026.35,
author = {Kopparty, Swastik and Ta-Shma, Amnon and Yakirevitch, Kedem},
title = {{Trace Hermitian Codes Have Vanishing Bias}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {35:1--35:28},
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.2026.35},
URN = {urn:nbn:de:0030-drops-270773},
doi = {10.4230/LIPIcs.CCC.2026.35},
annote = {Keywords: algebraic geometry codes, character sums, Stepanov method, error-correcting codes}
}
Document
Asymptotic Rank Speedup Theorems, Revisited
Abstract
Motivated by fast matrix multiplication and recent connections between asymptotic tensor rank and fine-grained complexity, we revisit classical tools from the matrix multiplication literature and develop a framework for obtaining improved asymptotic rank upper bounds for tensors beyond matrix multiplication.
In the 1980s, Coppersmith-Winograd and Strassen discovered a series of speedup theorems for asymptotic rank: in certain regimes, one can extract additional terms from a border rank upper bound on a tensor T, and then use these terms to obtain an improved asymptotic rank of T. We establish general speedup theorems that subsume these results and enable quantitative improvements. Two representative applications are:
1) The asymptotic rank of the small Coppersmith-Winograd tensor cw_q is less than its border rank. For instance, we prove ̰{R}(cw₂) < 3.931, improving on ̲{R}(cw₂) = 4. It is known that ̰{R}(cw₂) = 3 would imply ω = 2.
2) A general improvement over Strassen’s bound: we obtain an upper bound below d^{2ω/3} on the asymptotic rank of any d× d× d tensor. To make full use of speedups, we analyze degenerations in which both sides are nontrivial direct sums, a setting where the optimal quantitative bound one can achieve was previously unclear. We do so via an approach we call Strassen calculus: a systematic method for converting such degeneration data into explicit asymptotic rank bounds using Strassen’s theory of the asymptotic spectrum.
Cite as
Josh Alman and Baitian Li. Asymptotic Rank Speedup Theorems, Revisited. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 36:1-36:42, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{alman_et_al:LIPIcs.CCC.2026.36,
author = {Alman, Josh and Li, Baitian},
title = {{Asymptotic Rank Speedup Theorems, Revisited}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {36:1--36:42},
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.2026.36},
URN = {urn:nbn:de:0030-drops-270780},
doi = {10.4230/LIPIcs.CCC.2026.36},
annote = {Keywords: tensor rank, matrix multiplication, Coppersmith-Winograd tensor, asymptotic spectrum}
}
Document
Separations Above TFNP from Sherali-Adams Lower Bounds
Abstract
Unlike in TFNP, for which there is an abundance of problems capturing natural existence principles which are incomparable (in the black-box setting), Kleinberg et al. [Robert Kleinberg et al., 2021] observed that many of the natural problems considered so far in the second level of the total function polynomial hierarchy (TFΣ₂) reduce to the Strong Avoid problem. In this work, we prove that the Linear Ordering Principle does not reduce to Strong Avoid in the black-box setting, exhibiting the first TFΣ₂ problem that lies outside of the class of problems reducible to Strong Avoid.
The proof of our separation exploits a connection between total search problems in the polynomial hierarchy and proof complexity, recently developed by Fleming, Imrek, and Marciot [Fleming et al., 2025]. In particular, this implies that to show our separation, it suffices to show that there is no small proof of the Linear Ordering Principle in a Σ₂-variant of the Sherali-Adams proof system. To do so, we extend the classical pseudo-expectation method to the Σ₂ setting, showing that the existence of a Σ₂ pseudo-expectation precludes a Σ₂ Sherali-Adams proof. The main technical challenge is in proving the existence of such a pseudo-expectation, we manage to do so by solving a combinatorial covering problem about permutations. We also show that the extended pseudo-expectation bound implies that the Linear Ordering Principle cannot be reduced to any problem admitting a low-degree Sherali-Adams refutation.
Cite as
Noah Fleming, Anna Gál, Deniz Imrek, and Christophe Marciot. Separations Above TFNP from Sherali-Adams Lower Bounds. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 37:1-37:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fleming_et_al:LIPIcs.CCC.2026.37,
author = {Fleming, Noah and G\'{a}l, Anna and Imrek, Deniz and Marciot, Christophe},
title = {{Separations Above TFNP from Sherali-Adams Lower Bounds}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {37:1--37:23},
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.2026.37},
URN = {urn:nbn:de:0030-drops-270797},
doi = {10.4230/LIPIcs.CCC.2026.37},
annote = {Keywords: TFNP, Range Avoidance, Linear Ordering Principle, Separation, Sherali-Adams, Pseudo-Expectation}
}
Document
Non-Clifford Gates Are Required for Long-Term Memory
Abstract
We show that all Clifford circuits under interspersed depolarizing noise lose memory of their input exponentially quickly in the depth of the circuit, even when given access to a supply of fresh qubits initialized in arbitrary states. This result applies only in the absence of intermediate measurements, i.e. without intervention by a noiseless external observer such as a classical computer. Nonetheless, this result is surprising given that non-Clifford circuits with access to a supply of fresh qubits can preserve quantum states for arbitrary lengths of time, and even perform fault tolerant quantum computation on them, without requiring any intermediate measurements (Aharonov et al., STOC 1997). Our result shows that such fault tolerance protocols are impossible using only Clifford gates, demonstrating that non-Clifford gates are fundamentally required to store quantum or classical information for long periods of time.
Cite as
Jon Nelson, Joel Rajakumar, and Michael J. Gullans. Non-Clifford Gates Are Required for Long-Term Memory. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 38:1-38:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{nelson_et_al:LIPIcs.CCC.2026.38,
author = {Nelson, Jon and Rajakumar, Joel and Gullans, Michael J.},
title = {{Non-Clifford Gates Are Required for Long-Term Memory}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {38:1--38:13},
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.2026.38},
URN = {urn:nbn:de:0030-drops-270800},
doi = {10.4230/LIPIcs.CCC.2026.38},
annote = {Keywords: Fault Tolerance, Quantum Error Correction}
}
Document
Faux Determinism
Abstract
In the context of search problems, a pseudodeterministic algorithm is one which uses randomness but appears to be deterministic in its external behavior: it outputs the same answer when it is run on any one input multiple times. We introduce the relaxed notion of computationally-secure pseudodeterminism, which we call faux determinism. A faux-deterministic algorithm must appear to be deterministic to any external adversary with bounded resources. In other words, an algorithm is faux-deterministic if the task of "finding an input on which it fails to deterministically solve the search problem" is computationally intractable. In the black-box setting, we show that every verifiable search problem which has a randomized algorithm also has a faux-deterministic algorithm; this is not true of pseudodeterministic algorithms, which do not always exist. We highlight two reasons why this faux derandomization result is interesting: first, due to the computational indistinguishability, faux-deterministic algorithms may be used in place of pseudodeterministic ones in cryptographic settings. Second, from a conceptual point of view, our result helps explain why pseudodeterministic lower bounds are often hard to prove: such lower bounds must be inherently nonconstructive, since within FZPP∩FNP, there is no efficient black-box algorithm for finding a hard input for a given faux-deterministic algorithm.
Cite as
Shalev Ben-David and M. H. Ebtehaj. Faux Determinism. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 39:1-39:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bendavid_et_al:LIPIcs.CCC.2026.39,
author = {Ben-David, Shalev and Ebtehaj, M. H.},
title = {{Faux Determinism}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {39:1--39:25},
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.2026.39},
URN = {urn:nbn:de:0030-drops-270812},
doi = {10.4230/LIPIcs.CCC.2026.39},
annote = {Keywords: Decision Trees, Search Problems, Query Complexity, Pseudodeterminism, TFNP, Refutation}
}
Document
ETH-Hardness of Learning Monotone Circuits and Approximating Their Size
Abstract
We show the following hardness results for monotone learning and approximation of monotone circuit size:
1) Under the Randomised Exponential-Time Hypothesis (rETH), it requires time n^{Ω(log n)} to PAC-learn monotone formulas with n input bits and size s(n) = n by monotone circuits of size n^{(log n)^{1-ε}}, for every ε > 0.
2) Under the Randomised Exponential-Time Hypothesis (rETH), for any δ > 0, there is a polynomially bounded function m such that m^{1-δ}-multiplicatively approximating the minimum monotone circuit size of a monotone function consistent with a sequence of m(n) labelled examples {(x_i, b_i)} over n-bit inputs requires time m^{Ω(log(m))}. Our results are shown by a novel application of lifting arguments in proof and communication complexity to hardness of monotone learning, by building on the seminal result of Atserias and Müller [Atserias and Müller, 2020] on hardness of automating Resolution proofs.
Cite as
Bruno Cavalar, Susanna F. de Rezende, Matthew Gray, and Rahul Santhanam. ETH-Hardness of Learning Monotone Circuits and Approximating Their Size. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 40:1-40:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cavalar_et_al:LIPIcs.CCC.2026.40,
author = {Cavalar, Bruno and de Rezende, Susanna F. and Gray, Matthew and Santhanam, Rahul},
title = {{ETH-Hardness of Learning Monotone Circuits and Approximating Their Size}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {40:1--40:25},
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.2026.40},
URN = {urn:nbn:de:0030-drops-270826},
doi = {10.4230/LIPIcs.CCC.2026.40},
annote = {Keywords: monotone circuit complexity, PAC learning, lifting theorems, Resolution, meta-complexity, hardness of approximation, minimum circuit size problem}
}
Document
Systematic Data Structure Lower Bounds via the Query-With-Sketch Model
Abstract
We study data structure lower bounds for the Approximate Matrix Powering (AMP) problem. Given a substochastic, symmetric matrix 𝐌 ∈ ℝ^{n× n} and parameters k and α, the goal is to preprocess 𝐌 so as to answer entry queries (u,v)↦ 𝐌^{k}[u,v] up to additive error 1/n^{α}. We focus on AMP in the succinct and systematic regime, in which the data structure stores 𝐌 verbatim, uses an additional r bits of redundancy, and must answer queries by probing only a small number of entries of 𝐌.
Our main conceptual contribution is a general framework for proving probe-redundancy trade-offs for systematic data structures. We introduce the query-with-sketch model and develop a min-entropy-based approach that lifts conditional min-entropy bounds in the absence of redundancy to probe lower bounds in the presence of redundancy. We then establish these min-entropy bounds using problem-specific analytic and algebraic tools, for the downstream applications to AMP and its variants. As a consequence, our results provide new unconditional evidence toward a conjecture of Pătraşcu and Roditty on the space required for constant-time set-disjointness queries [Patrascu and Roditty, 2010].
Cite as
Sumegha Garg, Songhua He, Yuanzhi Li, Periklis A. Papakonstantinou, and Xin Yang. Systematic Data Structure Lower Bounds via the Query-With-Sketch Model. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 41:1-41:44, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{garg_et_al:LIPIcs.CCC.2026.41,
author = {Garg, Sumegha and He, Songhua and Li, Yuanzhi and Papakonstantinou, Periklis A. and Yang, Xin},
title = {{Systematic Data Structure Lower Bounds via the Query-With-Sketch Model}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {41:1--41:44},
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.2026.41},
URN = {urn:nbn:de:0030-drops-270833},
doi = {10.4230/LIPIcs.CCC.2026.41},
annote = {Keywords: systematic data structure, query-with-sketch model, approximate matrix powering}
}
Document
Randomized Separations in Black-Box TFNP
Abstract
We study the relationship between deterministic and randomized black-box reducibility between problems in TFNP. Our main contribution is a general technique that establishes equivalence between these reducibility types from specific TFNP problems to any TFNP problem. In particular, we show that this equivalence holds for reductions from complete problems in PPP, PPAD, PPA, and t-PPP. In turn, it strengthens all known black-box separations, originating from these classes, to randomized separations.
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Fedor Kiselev. Randomized Separations in Black-Box TFNP. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kiselev:LIPIcs.CCC.2026.42,
author = {Kiselev, Fedor},
title = {{Randomized Separations in Black-Box TFNP}},
booktitle = {41st Computational Complexity Conference (CCC 2026)},
pages = {42:1--42:17},
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.2026.42},
URN = {urn:nbn:de:0030-drops-270840},
doi = {10.4230/LIPIcs.CCC.2026.42},
annote = {Keywords: TFNP, Pigeonhole Principle}
}