DROPS

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DOI: 10.4230/LIPIcs.ITCS.2026.88

Lower Bounds and Separations for Torus Polynomials

Authors: Vaibhav Krishan and Sundar Vishwanathan

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

Abstract

The class ACC⁰ consists of Boolean functions that can be computed by constant-depth circuits of polynomial size with AND, NOT and MOD_m gates, where m is a natural number. At the frontier of our understanding lies a widely believed conjecture asserting that MAJORITY does not belong to ACC⁰. A few years ago, Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) introduced torus polynomial approximations as an approach towards this conjecture. Torus polynomials approximate Boolean functions when the fractional part of their value on Boolean points is close to half the value of the function. They reduced the conjecture that MAJORITY ∉ ACC⁰ to a conjecture concerning the non-existence of low degree torus polynomials that approximate MAJORITY. We reduce the non-existence problem further, to a statement about finding feasible solutions for an infinite family of linear programs. The main advantage of this statement is that it allows for incremental progress, which means finding feasible solutions for successively larger collections of these programs. As an immediate first step, we find feasible solutions for a large class of these linear programs, leaving only a finite set for further consideration. Our method is inspired by the method of dual polynomials, which is used to study the approximate degree of Boolean functions. Using our method, we also propose a way to progress further. We prove several additional key results with the same method, which include: - A lower bound on the degree of symmetric torus polynomials that approximate the AND function. As a consequence, we get a separation that symmetric torus polynomials are weaker than their asymmetric counterparts. - An error-degree trade-off for symmetric torus polynomials approximating the MAJORITY function, strengthening the corresponding result of Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019). - The first lower bounds against torus polynomials approximating AND, showcasing the power of the machinery we develop. This lower bound nearly matches the corresponding upper bound. Hence, we get an almost complete characterization of the torus polynomial approximation degree of AND. - Lower bounds against asymmetric torus polynomials approximating MAJORITY, or AND, in the very low error regime. This partially answers a question posed in Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) about error-reduction for torus polynomials.

Cite as

Vaibhav Krishan and Sundar Vishwanathan. Lower Bounds and Separations for Torus Polynomials. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 88:1-88:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{krishan_et_al:LIPIcs.ITCS.2026.88,
  author =	{Krishan, Vaibhav and Vishwanathan, Sundar},
  title =	{{Lower Bounds and Separations for Torus Polynomials}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{88:1--88:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.88},
  URN =		{urn:nbn:de:0030-drops-253751},
  doi =		{10.4230/LIPIcs.ITCS.2026.88},
  annote =	{Keywords: Circuit complexity, ACC, lower bounds, polynomials}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.58

Equality Is Far Weaker Than Constant-Cost Communication

Authors: Mika Göös, Nathaniel Harms, and Artur Riazanov

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

Abstract

We exhibit an n-bit communication problem with a constant-cost randomized protocol but which requires n^Ω(1) deterministic (or even non-deterministic) queries to an Equality oracle. Therefore, even constant-cost randomized protocols cannot be efficiently "derandomized" using Equality oracles. This improves on several recent results and answers a question from the survey of Hatami and Hatami (SIGACT News 2024). It also gives a significantly simpler and quantitatively superior proof of the main result of Fang, Göös, Harms, and Hatami (STOC 2025), that constant-cost communication does not reduce to the k-Hamming Distance hierarchy.

Cite as

Mika Göös, Nathaniel Harms, and Artur Riazanov. Equality Is Far Weaker Than Constant-Cost Communication. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{goos_et_al:LIPIcs.APPROX/RANDOM.2025.58,
  author =	{G\"{o}\"{o}s, Mika and Harms, Nathaniel and Riazanov, Artur},
  title =	{{Equality Is Far Weaker Than Constant-Cost Communication}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{58:1--58:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.58},
  URN =		{urn:nbn:de:0030-drops-244246},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.58},
  annote =	{Keywords: Equality oracle, constant-cost communication, gamma-2 norm, spectral norm}
}

Document

DOI: 10.4230/LIPIcs.CCC.2025.16

Direct Sums for Parity Decision Trees

Authors: Tyler Besselman, Mika Göös, Siyao Guo, Gilbert Maystre, and Weiqiang Yuan

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

Abstract

Direct sum theorems state that the cost of solving k instances of a problem is at least Ω(k) times the cost of solving a single instance. We prove the first such results in the randomised parity decision tree model. We show that a direct sum theorem holds whenever (1) the lower bound for parity decision trees is proved using the discrepancy method; or (2) the lower bound is proved relative to a product distribution.

Cite as

Tyler Besselman, Mika Göös, Siyao Guo, Gilbert Maystre, and Weiqiang Yuan. Direct Sums for Parity Decision Trees. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 16:1-16:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{besselman_et_al:LIPIcs.CCC.2025.16,
  author =	{Besselman, Tyler and G\"{o}\"{o}s, Mika and Guo, Siyao and Maystre, Gilbert and Yuan, Weiqiang},
  title =	{{Direct Sums for Parity Decision Trees}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{16:1--16:38},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.16},
  URN =		{urn:nbn:de:0030-drops-237105},
  doi =		{10.4230/LIPIcs.CCC.2025.16},
  annote =	{Keywords: direct sum, parity decision trees, query complexity}
}

Document

DOI: 10.4230/LIPIcs.CCC.2025.9

Pseudorandom Bits for Non-Commutative Programs

Authors: Chin Ho Lee and Emanuele Viola

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

Abstract

We obtain new explicit pseudorandom generators for several computational models involving groups. Our main results are as follows: 1) We consider read-once group-products over a finite group G, i.e., tests of the form ∏_{i=1}^n (g_i)^{x_i} where g_i ∈ G, a special case of read-once permutation branching programs. We give generators with optimal seed length c_G log(n/ε) over any p-group. The proof uses the small-bias plus noise paradigm, but derandomizes the noise to avoid the recursion in previous work. Our generator works when the bits are read in any order. Previously for any non-commutative group the best seed length was ≥ log n log(1/ε), even for a fixed order. 2) We give a reduction that "lifts" suitable generators for group products over G to a generator that fools width-w block products, i.e., tests of the form ∏ (g_i)^{f_i} where the f_i are arbitrary functions on disjoint blocks of w bits. Block products generalize several previously studied classes. The reduction applies to groups that are mixing in a representation-theoretic sense that we identify. 3) Combining (2) with (1) and other works we obtain new generators for block products over the quaternions or over any commutative group, with nearly optimal seed length. In particular, we obtain generators for read-once polynomials modulo any fixed m with nearly optimal seed length. Previously this was known only for m = 2. 4) We give a new generator for products over "mixing groups." The construction departs from previous work and uses representation theory. For constant error, we obtain optimal seed length, improving on previous work (which applied to any group). This paper identifies a challenge in the area that is reminiscent of a roadblock in circuit complexity - handling composite moduli - and points to several classes of groups to be attacked next.

Cite as

Chin Ho Lee and Emanuele Viola. Pseudorandom Bits for Non-Commutative Programs. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 9:1-9:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lee_et_al:LIPIcs.CCC.2025.9,
  author =	{Lee, Chin Ho and Viola, Emanuele},
  title =	{{Pseudorandom Bits for Non-Commutative Programs}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{9:1--9:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.9},
  URN =		{urn:nbn:de:0030-drops-237039},
  doi =		{10.4230/LIPIcs.CCC.2025.9},
  annote =	{Keywords: Group programs, Space-bounded derandomization, Representation theory}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.11

A Near-Optimal Polynomial Distance Lemma over Boolean Slices

Authors: Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, and Madhu Sudan

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

Abstract

The celebrated Ore-DeMillo-Lipton-Schwartz-Zippel (ODLSZ) lemma asserts that n-variate non-zero polynomial functions of degree d over a field 𝔽, are non-zero over any "grid" (points of the form Sⁿ for finite subset S ⊆ 𝔽) with probability at least max{|S|^{-d/(|S|-1)},1-d/|S|} over the choice of random point from the grid. In particular, over the Boolean cube (S = {0,1} ⊆ 𝔽), the lemma asserts non-zero polynomials are non-zero with probability at least 2^{-d}. In this work we extend the ODLSZ lemma optimally (up to lower-order terms) to "Boolean slices" i.e., points of Hamming weight exactly k. We show that non-zero polynomials on the slice are non-zero with probability (t/n)^{d}(1 - o_{n}(1)) where t = min{k,n-k} for every d ≤ k ≤ (n-d). As with the ODLSZ lemma, our results extend to polynomials over Abelian groups. This bound is tight upto the error term as evidenced by multilinear monomials of degree d, and it is also the case that some corrective term is necessary. A particularly interesting case is the "balanced slice" (k = n/2) where our lemma asserts that non-zero polynomials are non-zero with roughly the same probability on the slice as on the whole cube. The behaviour of low-degree polynomials over Boolean slices has received much attention in recent years. However, the problem of proving a tight version of the ODLSZ lemma does not seem to have been considered before, except for a recent work of Amireddy, Behera, Paraashar, Srinivasan and Sudan (SODA 2025), who established a sub-optimal bound of approximately ((k/n)⋅ (1-(k/n)))^d using a proof similar to that of the standard ODLSZ lemma. While the statement of our result mimics that of the ODLSZ lemma, our proof is significantly more intricate and involves spectral reasoning which is employed to show that a natural way of embedding a copy of the Boolean cube inside a balanced Boolean slice is a good sampler.

Cite as

Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, and Madhu Sudan. A Near-Optimal Polynomial Distance Lemma over Boolean Slices. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{amireddy_et_al:LIPIcs.ICALP.2025.11,
  author =	{Amireddy, Prashanth and Behera, Amik Raj and Srinivasan, Srikanth and Sudan, Madhu},
  title =	{{A Near-Optimal Polynomial Distance Lemma over Boolean Slices}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{11:1--11:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.11},
  URN =		{urn:nbn:de:0030-drops-233881},
  doi =		{10.4230/LIPIcs.ICALP.2025.11},
  annote =	{Keywords: Low-degree polynomials, Boolean slices, Schwartz-Zippel Lemma}
}

Document

DOI: 10.4230/LIPIcs.STACS.2025.58

Violating Constant Degree Hypothesis Requires Breaking Symmetry

Authors: Piotr Kawałek and Armin Weiß

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract

The Constant Degree Hypothesis was introduced by Barrington et. al. [David A. Mix Barrington et al., 1990] to study some extensions of q-groups by nilpotent groups and the power of these groups in a computation model called NuDFA (non-uniform DFA). In its simplest formulation, it establishes exponential lower bounds for MOD_q∘MOD_m∘AND_d circuits computing AND of unbounded arity n (for constant integers d,m and a prime q). While it has been proved in some special cases (including d = 1), it remains wide open in its general form for over 30 years. In this paper we prove that the hypothesis holds when we restrict our attention to symmetric circuits with m being a prime. While we build upon techniques by Grolmusz and Tardos [Vince Grolmusz and Gábor Tardos, 2000], we have to prove a new symmetric version of their Degree Decreasing Lemma and use it to simplify circuits in a symmetry-preserving way. Moreover, to establish the result, we perform a careful analysis of automorphism groups of MOD_m∘AND_d subcircuits and study the periodic behaviour of the computed functions. Our methods also yield lower bounds when d is treated as a function of n. Finally, we present a construction of symmetric MOD_q∘MOD_m∘AND_d circuits that almost matches our lower bound and conclude that a symmetric function f can be computed by symmetric MOD_q∘MOD_p∘AND_d circuits of quasipolynomial size if and only if f has periods of polylogarithmic length of the form p^k q^𝓁.

Cite as

Piotr Kawałek and Armin Weiß. Violating Constant Degree Hypothesis Requires Breaking Symmetry. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 58:1-58:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kawalek_et_al:LIPIcs.STACS.2025.58,
  author =	{Kawa{\l}ek, Piotr and Wei{\ss}, Armin},
  title =	{{Violating Constant Degree Hypothesis Requires Breaking Symmetry}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{58:1--58:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.58},
  URN =		{urn:nbn:de:0030-drops-228837},
  doi =		{10.4230/LIPIcs.STACS.2025.58},
  annote =	{Keywords: Circuit lower bounds, constant degree hypothesis, permutation groups, CC⁰-circuits}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2025.37

A Lower Bound on the Trace Norm of Boolean Matrices and Its Applications

Authors: Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, Aleksandar Nikolov, Toniann Pitassi, and Morgan Shirley

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

We present a simple method based on a variant of Hölder’s inequality to lower-bound the trace norm of Boolean matrices. As the main result, we obtain an exponential separation between the randomized decision tree depth and the spectral norm (i.e. the Fourier L₁-norm) of a Boolean function. This answers an open question of Cheung, Hatami, Hosseini and Shirley (CCC 2023). As immediate consequences, we obtain the following results. - We give an exponential separation between the logarithm of the randomized and the deterministic parity decision tree size. This is in sharp contrast with the standard binary decision tree setting where the logarithms of randomized and deterministic decision tree size are essentially polynomially related, as shown recently by Chattopadhyay, Dahiya, Mande, Radhakrishnan, and Sanyal (STOC 2023). - We give an exponential separation between the approximate and the exact spectral norm for Boolean functions. - We give an exponential separation for XOR functions between the deterministic communication complexity with oracle access to Equality function (D^EQ) and randomized communication complexity. Previously, such a separation was known for general Boolean matrices by Chattopadhyay, Lovett, and Vinyals (CCC 2019) using the Integer Inner Product (IIP) function. - Finally, our method gives an elementary and short proof for the mentioned exponential D^EQ lower bound of Chattopadhyay, Lovett, and Vinyals for Integer Inner Product (IIP).

Cite as

Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, Aleksandar Nikolov, Toniann Pitassi, and Morgan Shirley. A Lower Bound on the Trace Norm of Boolean Matrices and Its Applications. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cheung_et_al:LIPIcs.ITCS.2025.37,
  author =	{Cheung, Tsun-Ming and Hatami, Hamed and Hosseini, Kaave and Nikolov, Aleksandar and Pitassi, Toniann and Shirley, Morgan},
  title =	{{A Lower Bound on the Trace Norm of Boolean Matrices and Its Applications}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{37:1--37:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.37},
  URN =		{urn:nbn:de:0030-drops-226654},
  doi =		{10.4230/LIPIcs.ITCS.2025.37},
  annote =	{Keywords: Boolean function complexity, parity decision trees, randomized communication complexity}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2025.78

Randomized Lifting to Semi-Structured Communication Complexity via Linear Diversity

Authors: Vladimir Podolskii and Alexander Shekhovtsov

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

We study query-to-communication lifting. The major open problem in this area is to prove a lifting theorem for gadgets of constant size. The recent paper [Paul Beame and Sajin Koroth, 2023] introduces semi-structured communication complexity, in which one of the players can only send parities of their input bits. They have shown that for any m ≥ 4 deterministic decision tree complexity of a function f can be lifted to the so called semi-structured communication complexity of f∘Ind_m, where Ind_m is the Indexing gadget. As our main contribution we extend these results to randomized setting. Our results also apply to a substantially larger set of gadgets. More specifically, we introduce a new complexity measure of gadgets, linear diversity. For all gadgets g with non-trivial linear diversity we show that randomized decision tree complexity of f lifts to randomized semi-structured communication complexity of f∘g. In particular, this gives tight lifting results for Indexing gadget Ind_m, Inner Product gadget IP_m for all m ≥ 2, and for Majority gadget MAJ_m for all m ≥ 4. We prove the same results for deterministic case. From our result it immediately follows that deterministic/randomized decision tree complexity lifts to deterministic/randomized parity decision tree complexity. For randomized case this is the first result of this type. For deterministic case, our result improves the bound in [Arkadev Chattopadhyay et al., 2023] for Inner Product gadget. To obtain our results we introduce a new secret sets approach to simulation of semi-structured communication protocols by decision trees. It allows us to simulate (restricted classes of) communication protocols on truly uniform distribution of inputs.

Cite as

Vladimir Podolskii and Alexander Shekhovtsov. Randomized Lifting to Semi-Structured Communication Complexity via Linear Diversity. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 78:1-78:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{podolskii_et_al:LIPIcs.ITCS.2025.78,
  author =	{Podolskii, Vladimir and Shekhovtsov, Alexander},
  title =	{{Randomized Lifting to Semi-Structured Communication Complexity via Linear Diversity}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{78:1--78:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.78},
  URN =		{urn:nbn:de:0030-drops-227061},
  doi =		{10.4230/LIPIcs.ITCS.2025.78},
  annote =	{Keywords: communication complexity, decision trees, lifting}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.82

Refuting Approaches to the Log-Rank Conjecture for XOR Functions

Authors: Hamed Hatami, Kaave Hosseini, Shachar Lovett, and Anthony Ostuni

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

Abstract

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang, Wong, Xie, and Zhang (FOCS'13), and the second is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

Cite as

Hamed Hatami, Kaave Hosseini, Shachar Lovett, and Anthony Ostuni. Refuting Approaches to the Log-Rank Conjecture for XOR Functions. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 82:1-82:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hatami_et_al:LIPIcs.ICALP.2024.82,
  author =	{Hatami, Hamed and Hosseini, Kaave and Lovett, Shachar and Ostuni, Anthony},
  title =	{{Refuting Approaches to the Log-Rank Conjecture for XOR Functions}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{82:1--82:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.82},
  URN =		{urn:nbn:de:0030-drops-202252},
  doi =		{10.4230/LIPIcs.ICALP.2024.82},
  annote =	{Keywords: Communication complexity, log-rank conjecture, XOR functions, additive structure}
}

Document

DOI: 10.4230/LIPIcs.CCC.2023.1

Separation of the Factorization Norm and Randomized Communication Complexity

Authors: Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, and Morgan Shirley

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract

In an influential paper, Linial and Shraibman (STOC '07) introduced the factorization norm as a powerful tool for proving lower bounds against randomized and quantum communication complexities. They showed that the logarithm of the approximate γ₂-factorization norm is a lower bound for these parameters and asked whether a stronger lower bound that replaces approximate γ₂ norm with the γ₂ norm holds. We answer the question of Linial and Shraibman in the negative by exhibiting a 2ⁿ×2ⁿ Boolean matrix with γ₂ norm 2^Ω(n) and randomized communication complexity O(log n). As a corollary, we recover the recent result of Chattopadhyay, Lovett, and Vinyals (CCC '19) that deterministic protocols with access to an Equality oracle are exponentially weaker than (one-sided error) randomized protocols. In fact, as a stronger consequence, our result implies an exponential separation between the power of unambiguous nondeterministic protocols with access to Equality oracle and (one-sided error) randomized protocols, which answers a question of Pitassi, Shirley, and Shraibman (ITSC '23). Our result also implies a conjecture of Sherif (Ph.D. thesis) that the γ₂ norm of the Integer Inner Product function (IIP) in dimension 3 or higher is exponential in its input size.

Cite as

Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, and Morgan Shirley. Separation of the Factorization Norm and Randomized Communication Complexity. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cheung_et_al:LIPIcs.CCC.2023.1,
  author =	{Cheung, Tsun-Ming and Hatami, Hamed and Hosseini, Kaave and Shirley, Morgan},
  title =	{{Separation of the Factorization Norm and Randomized Communication Complexity}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{1:1--1:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.1},
  URN =		{urn:nbn:de:0030-drops-182714},
  doi =		{10.4230/LIPIcs.CCC.2023.1},
  annote =	{Keywords: Factorization norms, randomized communication complexity}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2023.42

Online Learning and Disambiguations of Partial Concept Classes

Authors: Tsun-Ming Cheung, Hamed Hatami, Pooya Hatami, and Kaave Hosseini

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract

In a recent article, Alon, Hanneke, Holzman, and Moran (FOCS '21) introduced a unifying framework to study the learnability of classes of partial concepts. One of the central questions studied in their work is whether the learnability of a partial concept class is always inherited from the learnability of some "extension" of it to a total concept class. They showed this is not the case for PAC learning but left the problem open for the stronger notion of online learnability. We resolve this problem by constructing a class of partial concepts that is online learnable, but no extension of it to a class of total concepts is online learnable (or even PAC learnable).

Cite as

Tsun-Ming Cheung, Hamed Hatami, Pooya Hatami, and Kaave Hosseini. Online Learning and Disambiguations of Partial Concept Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cheung_et_al:LIPIcs.ICALP.2023.42,
  author =	{Cheung, Tsun-Ming and Hatami, Hamed and Hatami, Pooya and Hosseini, Kaave},
  title =	{{Online Learning and Disambiguations of Partial Concept Classes}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{42:1--42:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.42},
  URN =		{urn:nbn:de:0030-drops-180946},
  doi =		{10.4230/LIPIcs.ICALP.2023.42},
  annote =	{Keywords: Online learning, Littlestone dimension, VC dimension, partial concept class, clique vs independent set, Alon-Saks-Seymour conjecture, Standard Optimal Algorithm, PAC learning}
}

@InProceedings{cheung_et_al:LIPIcs.ICALP.2023.42,
  author =	{Cheung, Tsun-Ming and Hatami, Hamed and Hatami, Pooya and Hosseini, Kaave},
  title =	{{Online Learning and Disambiguations of Partial Concept Classes}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{42:1--42:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.42},
  URN =		{urn:nbn:de:0030-drops-180946},
  doi =		{10.4230/LIPIcs.ICALP.2023.42},
  annote =	{Keywords: Online learning, Littlestone dimension, VC dimension, partial concept class, clique vs independent set, Alon-Saks-Seymour conjecture, Standard Optimal Algorithm, PAC learning}
}

Document

DOI: 10.4230/LIPIcs.CCC.2022.13

A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates

Authors: Ivan Mihajlin and Anastasia Sofronova

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

Abstract

We prove that a modification of Andreev’s function is not computable by (3 + α - ε) log(n) depth De Morgan formula with (2α - ε)log{n} layers of AND gates at the top for any 0 < α < 1/5 and any constant ε > 0. In order to do this, we prove a weak variant of Karchmer-Raz-Wigderson conjecture. To be more precise, we prove the existence of two functions f : {0,1}ⁿ → {0,1} and g : {0,1}ⁿ → {0,1}ⁿ such that f(g(x) ⊕ y) is not computable by depth (1 + α - ε) n formulas with (2 α - ε) n layers of AND gates at the top. We do this by a top-down approach, which was only used before for depth-3 model. Our technical contribution includes combinatorial insights into structure of composition with random boolean function, which led us to introducing a notion of well-mixed sets. A set of functions is well-mixed if, when composed with a random function, it does not have subsets that agree on large fractions of inputs. We use probabilistic method to prove the existence of well-mixed sets.

Cite as

Ivan Mihajlin and Anastasia Sofronova. A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mihajlin_et_al:LIPIcs.CCC.2022.13,
  author =	{Mihajlin, Ivan and Sofronova, Anastasia},
  title =	{{A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{13:1--13:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.13},
  URN =		{urn:nbn:de:0030-drops-165755},
  doi =		{10.4230/LIPIcs.CCC.2022.13},
  annote =	{Keywords: formula complexity, communication complexity, Karchmer-Raz-Wigderson conjecture, De Morgan formulas}
}

Document

DOI: 10.4230/LIPIcs.CCC.2020.18

Sign Rank vs Discrepancy

Authors: Hamed Hatami, Kaave Hosseini, and Shachar Lovett

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

Abstract

Sign-rank and discrepancy are two central notions in communication complexity. The seminal work of Babai, Frankl, and Simon from 1986 initiated an active line of research that investigates the gap between these two notions. In this article, we establish the strongest possible separation by constructing a boolean matrix whose sign-rank is only 3, and yet its discrepancy is 2^{-Ω(n)}. We note that every matrix of sign-rank 2 has discrepancy n^{-O(1)}. Our result in particular implies that there are boolean functions with O(1) unbounded error randomized communication complexity while having Ω(n) weakly unbounded error randomized communication complexity.

Cite as

Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Sign Rank vs Discrepancy. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hatami_et_al:LIPIcs.CCC.2020.18,
  author =	{Hatami, Hamed and Hosseini, Kaave and Lovett, Shachar},
  title =	{{Sign Rank vs Discrepancy}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{18:1--18:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.18},
  URN =		{urn:nbn:de:0030-drops-125700},
  doi =		{10.4230/LIPIcs.CCC.2020.18},
  annote =	{Keywords: Discrepancy, sign rank, Unbounded-error communication complexity, weakly unbounded error communication complexity}
}

Document

DOI: 10.4230/LIPIcs.CCC.2019.13

Optimality of Linear Sketching Under Modular Updates

Authors: Kaave Hosseini, Shachar Lovett, and Grigory Yaroslavtsev

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract

We study the relation between streaming algorithms and linear sketching algorithms, in the context of binary updates. We show that for inputs in n dimensions, the existence of efficient streaming algorithms which can process Omega(n^2) updates implies efficient linear sketching algorithms with comparable cost. This improves upon the previous work of Li, Nguyen and Woodruff [Yi Li et al., 2014] and Ai, Hu, Li and Woodruff [Yuqing Ai et al., 2016] which required a triple-exponential number of updates to achieve a similar result for updates over integers. We extend our results to updates modulo p for integers p >= 2, and to approximation instead of exact computation.

Cite as

Kaave Hosseini, Shachar Lovett, and Grigory Yaroslavtsev. Optimality of Linear Sketching Under Modular Updates. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hosseini_et_al:LIPIcs.CCC.2019.13,
  author =	{Hosseini, Kaave and Lovett, Shachar and Yaroslavtsev, Grigory},
  title =	{{Optimality of Linear Sketching Under Modular Updates}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{13:1--13:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.13},
  URN =		{urn:nbn:de:0030-drops-108355},
  doi =		{10.4230/LIPIcs.CCC.2019.13},
  annote =	{Keywords: communication complexity, linear sketching, streaming algorithm}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2019.13

Torus Polynomials: An Algebraic Approach to ACC Lower Bounds

Authors: Abhishek Bhrushundi, Kaave Hosseini, Shachar Lovett, and Sankeerth Rao

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract

We propose an algebraic approach to proving circuit lower bounds for ACC^0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC^0 and ACC^0 can be reformulated in this framework, implying that ACC^0 can be approximated by low-degree torus polynomials. Furthermore, as a step towards proving ACC^0 lower bounds for the majority function via our approach, we show that MAJORITY cannot be approximated by low-degree symmetric torus polynomials. We also pose several open problems related to our framework.

Cite as

Abhishek Bhrushundi, Kaave Hosseini, Shachar Lovett, and Sankeerth Rao. Torus Polynomials: An Algebraic Approach to ACC Lower Bounds. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bhrushundi_et_al:LIPIcs.ITCS.2019.13,
  author =	{Bhrushundi, Abhishek and Hosseini, Kaave and Lovett, Shachar and Rao, Sankeerth},
  title =	{{Torus Polynomials: An Algebraic Approach to ACC Lower Bounds}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.13},
  URN =		{urn:nbn:de:0030-drops-101066},
  doi =		{10.4230/LIPIcs.ITCS.2019.13},
  annote =	{Keywords: Circuit complexity, ACC, lower bounds, polynomials}
}

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