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DOI: 10.4230/LIPIcs.ITCS.2026.9

On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

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

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

Abstract

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

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

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@InProceedings{andrews_et_al:LIPIcs.ITCS.2026.9,
  author =	{Andrews, Robert and Armand, Jules and Dwivedi, Prateek and Hansen, Magnus Rahbek Dalgaard and Limaye, Nutan and Srinivasan, Srikanth and Tavenas, S\'{e}bastien},
  title =	{{On Closure Properties of Read-Once Oblivious Algebraic Branching Programs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{9:1--9:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.9},
  URN =		{urn:nbn:de:0030-drops-252964},
  doi =		{10.4230/LIPIcs.ITCS.2026.9},
  annote =	{Keywords: Factoring, Closure Properties, Sparsity Bounds, Symmetric Polynomials, roABP, Expander Graphs}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2025.19

Monotone Bounded-Depth Complexity of Homomorphism Polynomials

Authors: C.S. Bhargav, Shiteng Chen, Radu Curticapean, and Prateek Dwivedi

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

For every fixed graph H, it is known that homomorphism counts from H and colorful H-subgraph counts can be determined in O(n^{t+1}) time on n-vertex input graphs G, where t is the treewidth of H. On the other hand, a running time of n^{o(t / log t)} would refute the exponential-time hypothesis. Komarath, Pandey, and Rahul (Algorithmica, 2023) studied algebraic variants of these counting problems, i.e., homomorphism and subgraph polynomials for fixed graphs H. These polynomials are weighted sums over the objects counted above, where each object is weighted by the product of variables corresponding to edges contained in the object. As shown by Komarath et al., the monotone circuit complexity of the homomorphism polynomial for H is Θ(n^{tw(H)+1}). In this paper, we characterize the power of monotone bounded-depth circuits for homomorphism and colorful subgraph polynomials. This leads us to discover a natural hierarchy of graph parameters tw_Δ(H), for fixed Δ ∈ ℕ, which capture the width of tree-decompositions for H when the underlying tree is required to have depth at most Δ. We prove that monotone circuits of product-depth Δ computing the homomorphism polynomial for H require size Θ(n^{tw_Δ(H^{†})+1}), where H^{†} is the graph obtained from H by removing all degree-1 vertices. This allows us to derive an optimal depth hierarchy theorem for monotone bounded-depth circuits through graph-theoretic arguments.

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C.S. Bhargav, Shiteng Chen, Radu Curticapean, and Prateek Dwivedi. Monotone Bounded-Depth Complexity of Homomorphism Polynomials. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhargav_et_al:LIPIcs.MFCS.2025.19,
  author =	{Bhargav, C.S. and Chen, Shiteng and Curticapean, Radu and Dwivedi, Prateek},
  title =	{{Monotone Bounded-Depth Complexity of Homomorphism Polynomials}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{19:1--19:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.19},
  URN =		{urn:nbn:de:0030-drops-241269},
  doi =		{10.4230/LIPIcs.MFCS.2025.19},
  annote =	{Keywords: algebraic complexity, homomorphisms, monotone circuit complexity, bounded-depth circuits, treewidth, pathwidth}
}

Document

DOI: 10.4230/LIPIcs.CCC.2021.11

Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits

Authors: Pranjal Dutta, Prateek Dwivedi, and Nitin Saxena

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract

Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-4 reduction (Agrawal & Vinay, FOCS'08) has made PIT for depth-4 circuits, an enticing pursuit. The largely open special-cases of sum-product-of-sum-of-univariates (Σ^[k] Π Σ ∧) and sum-product-of-constant-degree-polynomials (Σ^[k] Π Σ Π^[δ]), for constants k, δ, have been a source of many great ideas in the last two decades. For eg. depth-3 ideas (Dvir & Shpilka, STOC'05; Kayal & Saxena, CCC'06; Saxena & Seshadhri, FOCS'10, STOC'11); depth-4 ideas (Beecken, Mittmann & Saxena, ICALP'11; Saha,Saxena & Saptharishi, Comput.Compl.'13; Forbes, FOCS'15; Kumar & Saraf, CCC'16); geometric Sylvester-Gallai ideas (Kayal & Saraf, FOCS'09; Shpilka, STOC'19; Peleg & Shpilka, CCC'20, STOC'21). We solve two of the basic underlying open problems in this work. We give the first polynomial-time PIT for Σ^[k] Π Σ ∧. Further, we give the first quasipolynomial time blackbox PIT for both Σ^[k] Π Σ ∧ and Σ^[k] Π Σ Π^[δ]. No subexponential time algorithm was known prior to this work (even if k = δ = 3). A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top Π-gate to ∧.

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Pranjal Dutta, Prateek Dwivedi, and Nitin Saxena. Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 11:1-11:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dutta_et_al:LIPIcs.CCC.2021.11,
  author =	{Dutta, Pranjal and Dwivedi, Prateek and Saxena, Nitin},
  title =	{{Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{11:1--11:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.11},
  URN =		{urn:nbn:de:0030-drops-142857},
  doi =		{10.4230/LIPIcs.CCC.2021.11},
  annote =	{Keywords: Polynomial identity testing, hitting set, depth-4 circuits}
}

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Documents authored by Dwivedi, Prateek

On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

Abstract

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Monotone Bounded-Depth Complexity of Homomorphism Polynomials

Abstract

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Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits

Abstract

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