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Documents authored by Behera, Amik Raj

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RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.34
Eigenvalue Bounds for Symmetric Markov Chains on Multislices with Applications

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

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

Abstract
We consider random walks on "balanced multislices" of any "grid" that respects the "symmetries" of the grid, and show that a broad class of such walks are good spectral expanders. (A grid is a set of points of the form 𝒮ⁿ for finite 𝒮, and a balanced multi-slice is the subset that contains an equal number of coordinates taking every value in 𝒮. A walk respects symmetries if the probability of going from u = (u_1,…,u_n) to v = (v_1,…,v_n) is invariant under simultaneous permutations of the coordinates of u and v.) Our main theorem shows that, under some technical conditions, every such walk where a single step leads to an almost 𝒪(1)-wise independent distribution on the next state, conditioned on the previous state, satisfies a non-trivially small singular value bound. We give two applications of our theorem to error-correcting codes: (1) We give an analog of the Ore-DeMillo-Lipton-Schwartz-Zippel lemma for polynomials, and junta-sums, over balanced multislices. (2) We also give a local list-correction algorithm for d-junta-sums mapping an arbitrary grid 𝒮ⁿ to an Abelian group, correcting from a near-optimal (1/|𝒮|^d - ε) fraction of errors for every ε > 0, where a d-junta-sum is a sum of (arbitrarily many) d-juntas (and a d-junta is a function that depends on only d of the n variables). Our proofs are obtained by exploring the representation theory of the symmetric group and merging it with some careful spectral analysis.
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Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, and Madhu Sudan. Eigenvalue Bounds for Symmetric Markov Chains on Multislices with Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 34:1-34:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{amireddy_et_al:LIPIcs.APPROX/RANDOM.2025.34,
  author =	{Amireddy, Prashanth and Behera, Amik Raj and Srinivasan, Srikanth and Sudan, Madhu},
  title =	{{Eigenvalue Bounds for Symmetric Markov Chains on Multislices with Applications}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{34:1--34:12},
  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.2025.34},
  URN =		{urn:nbn:de:0030-drops-244004},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.34},
  annote =	{Keywords: Markov Chains, Random Walk, Multislices, Representation Theory of Symmetric Group, Local Correction, Low-degree Polynomials, Polynomial Distance Lemma}
}
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.
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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
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.22
New Bounds for the Ideal Proof System in Positive Characteristic

Authors: Amik Raj Behera, Nutan Limaye, Varun Ramanathan, and Srikanth Srinivasan

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

Abstract
In this work, we prove upper and lower bounds over fields of positive characteristics for several fragments of the Ideal Proof System (IPS), an algebraic proof system introduced by Grochow and Pitassi (J. ACM 2018). Our results extend the works of Forbes, Shpilka, Tzameret, and Wigderson (Theory of Computing 2021) and also of Govindasamy, Hakoniemi, and Tzameret (FOCS 2022). These works primarily focused on proof systems over fields of characteristic 0, and we are able to extend these results to positive characteristic. The question of proving general IPS lower bounds over positive characteristic is motivated by the important question of proving AC⁰[p]-Frege lower bounds. This connection was observed by Grochow and Pitassi (J. ACM 2018). Additional motivation comes from recent developments in algebraic complexity theory due to Forbes (CCC 2024) who showed how to extend previous lower bounds over characteristic 0 to positive characteristic. In our work, we adapt the functional lower bound method of Forbes et al. (Theory of Computing 2021) to prove exponential-size lower bounds for various subsystems of IPS. In order to establish these size lower bounds, we first prove a tight degree lower bound for a variant of Subset Sum over positive characteristic. This forms the core of all our lower bounds. Additionally, we derive upper bounds for the instances presented above. We show that they have efficient constant-depth IPS refutations. This demonstrates that constant-depth IPS refutations are stronger than the proof systems considered above even in positive characteristic. We also show that constant-depth IPS can efficiently refute a general class of instances, namely all symmetric instances, thereby further uncovering the strength of these algebraic proofs in positive characteristic.
Cite as

Amik Raj Behera, Nutan Limaye, Varun Ramanathan, and Srikanth Srinivasan. New Bounds for the Ideal Proof System in Positive Characteristic. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{behera_et_al:LIPIcs.ICALP.2025.22,
  author =	{Behera, Amik Raj and Limaye, Nutan and Ramanathan, Varun and Srinivasan, Srikanth},
  title =	{{New Bounds for the Ideal Proof System in Positive Characteristic}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{22:1--22:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.22},
  URN =		{urn:nbn:de:0030-drops-233992},
  doi =		{10.4230/LIPIcs.ICALP.2025.22},
  annote =	{Keywords: Ideal Proof Systems, Algebraic Complexity, Positive Characteristic}
}
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