迭代Chvatal-Gomory割与数的几何
Iterated Chvatal-Gomory Cuts and the Geometry of Numbers
摘要：
Chvatal-Gomory割平面（简称CG割）是整数规划的基本工具。给定任何一个CG-割，通过对其乘子向量进行模1的“迭代”，可以导出CG-割的整个族。这自然会引出两个问题：第一，哪些迭代对应于最强的切割，第二，我们能有效地找到这样的强切割吗？我们回答了第一个问题的经验，通过显示，一个特定的方法来选择最佳往往比其他几个更好。该方法的本质在于解决一个非线性最优化问题在一个特殊的格与CG-割。然后，我们提供了第二个问题的部分答案，通过提出一个多项式时间算法，产生一个强的在一定的定义明确的意义上的时间序列。该算法是基于从算法的数的几何的结果。
英文摘要：
Chvatal-Gomory cutting planes (CG-cuts for short) are a fundamental tool in Integer Programming. Given any single CG-cut, one can derive an entire family of CG-cuts, by `iterating' its multiplier vector modulo one. This leads naturally to two questions: first, which iterates correspond to the strongest cuts, and, second, can we find such strong cuts efficiently? We answer the first question empirically, by showing that one specific approach for selecting the iterate tends to perform much better than several others. The approach essentially consists in solving a nonlinear optimization problem over a special lattice associated with the CG-cut. We then provide a partial answer to the second question, by presenting a polynomial-time algorithm that yields an iterate that is strong in a certain well-defined sense. The algorithm is based on results from the algorithmic geometry of numbers.
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