  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.
