  For several decades the dominant techniques for integer linear programming
have been branching and cutting planes. Recently, several authors have
developed core point methods for solving symmetric integer linear programs
(ILPs). An integer point is called a core point if its orbit polytope is
lattice-free. It has been shown that for symmetric ILPs, optimizing over the
set of core points gives the same answer as considering the entire space.
Existing core point techniques rely on the number of core points (or
equivalence classes) being finite, which requires special symmetry groups. In
this paper we develop some new methods for solving symmetric ILPs (based on
outer approximations of core points) that do not depend on finiteness but are
more efficient if the group has large disjoint cycles in its set of generators.
