Polynomial time algorithms in invariant theory for torus actions

Peter Bürgisser*, M. Levent Doğan*, Visu Makam*, Michael Walter*, Avi Wigderson*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations

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.

Original languageEnglish
Title of host publication36th Computational Complexity Conference, CCC 2021
EditorsValentine Kabanets
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771931
DOIs
StatePublished - 1 Jul 2021
Externally publishedYes
Event36th Computational Complexity Conference, CCC 2021 - Virtual, Toronto, Canada
Duration: 20 Jul 202123 Jul 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume200
ISSN (Print)1868-8969

Conference

Conference36th Computational Complexity Conference, CCC 2021
Country/TerritoryCanada
CityVirtual, Toronto
Period20/07/2123/07/21

Bibliographical note

Publisher Copyright:
© Peter Bürgisser, M. Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson;

Keywords

  • Computational invariant theory
  • Geometric complexity theory
  • Orbit closure intersection problem

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