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An action of a group on a vector space partitions the latter into a set oforbits. 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 ofproblems within mathematics, physics and computer science, optimization andstatistics. These orbit problems extend the more basic null cone problem, whosealgorithmic complexity has seen significant progress in recent years.  In this paper, we initiate a study of these problems by focusing on theactions of commutative groups (namely, tori). We explain how this setting ismotivated from questions in algebraic complexity, and is still rich enough tocapture interesting combinatorial algorithmic problems. While the structuraltheory of commutative actions is well understood, no general efficientalgorithms were known for the aforementioned problems. Our main results arepolynomial time algorithms for all three problems. We also show how toefficiently find separating invariants for orbits, and how to compute systemsof generating rational invariants for these actions (in contrast, forpolynomial invariants the latter is known to be hard). Our techniques are basedon a combination of fundamental results in invariant theory, linearprogramming, and algorithmic lattice theory.

Polynomial time algorithms in invariant theory for torus actions