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DOI: 10.4230/LIPIcs.CCC.2026.32

Fixed-Parameter Degree Bounds and Complexity of the Orbit Closure Intersection Problem for Tensors

Authors: M. Levent Doğan, John Maar, Rafael Oliveira, and Youming Qiao

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

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.

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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

DOI: 10.4230/LIPIcs.CCC.2021.32

Polynomial Time Algorithms in Invariant Theory for Torus Actions

Authors: Peter Bürgisser, M. Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson

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

Abstract

An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years. In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory.

Cite as

Peter Bürgisser, M. Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson. Polynomial Time Algorithms in Invariant Theory for Torus Actions. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 32:1-32:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{burgisser_et_al:LIPIcs.CCC.2021.32,
  author =	{B\"{u}rgisser, Peter and Do\u{g}an, M. Levent and Makam, Visu and Walter, Michael and Wigderson, Avi},
  title =	{{Polynomial Time Algorithms in Invariant Theory for Torus Actions}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{32:1--32:30},
  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.2021.32},
  URN =		{urn:nbn:de:0030-drops-143062},
  doi =		{10.4230/LIPIcs.CCC.2021.32},
  annote =	{Keywords: computational invariant theory, geometric complexity theory, orbit closure intersection problem}
}

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Documents authored by Doğan, M. Levent

Fixed-Parameter Degree Bounds and Complexity of the Orbit Closure Intersection Problem for Tensors

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Polynomial Time Algorithms in Invariant Theory for Torus Actions

Abstract

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