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 language | English |
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Title of host publication | 36th Computational Complexity Conference, CCC 2021 |
Editors | Valentine Kabanets |
Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
ISBN (Electronic) | 9783959771931 |
DOIs | |
State | Published - 1 Jul 2021 |
Externally published | Yes |
Event | 36th Computational Complexity Conference, CCC 2021 - Virtual, Toronto, Canada Duration: 20 Jul 2021 → 23 Jul 2021 |
Publication series
Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 200 |
ISSN (Print) | 1868-8969 |
Conference
Conference | 36th Computational Complexity Conference, CCC 2021 |
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Country/Territory | Canada |
City | Virtual, Toronto |
Period | 20/07/21 → 23/07/21 |
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