Problem
Research questions and friction points this paper is trying to address.
Develops methods for symmetric integer linear programs
Uses outer approximations of core points
Improves efficiency for large symmetry groups
Outer approximations of core points for integer programming
🤖 AI Summary
Traditional core point methods for symmetric integer linear programming (ILP) rely on the restrictive “finiteness of core points” assumption—requiring the symmetry group to exhibit a specific orbit structure—severely limiting their applicability.
Method: We propose a novel outer-approximation paradigm that dispenses with this assumption. Our approach introduces a general core-point outer-approximation framework, integrating orbit polytope analysis under group actions with convex-hull outer-approximation techniques; leverages large disjoint cycles in group generators to accelerate computation; and equivalently reduces the original problem to a geometrically tighter approximate core-point set.
Contribution/Results: We rigorously prove convergence and optimality preservation of the outer-approximation sequence. The method extends core-point theory to arbitrary finite symmetry groups, significantly broadening applicability. It also improves computational efficiency for group actions containing large disjoint cycles. Overall, it provides a theoretically sound and computationally viable new pathway for solving symmetric ILPs.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Outer approximations of core points
Solving symmetric ILPs efficiently
Handling large disjoint cycles
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