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.

翻译：数十年来，整数线性规划的主流技术一直是分支定界与割平面法。近期，多位学者提出了求解对称整数线性规划的核心点方法。若某整数点的轨道多胞体是无晶格的，则该点被称为核心点。研究表明，对于对称整数线性规划，在核心点集上优化所得结果与在全空间上优化等价。现有核心点技术依赖于核心点（或其等价类）数目有限这一前提，这需要特殊的对称群结构。本文提出若干基于核心点外部逼近的对称整数线性规划新解法，这些方法不依赖有限性假设，但当群的生成元集包含较大不交环时具有更高效率。