基于聚集的整数规划包装与覆盖割平面
Aggregation-based cutting-planes for packing and covering integer programs
摘要：
在本文中，我们研究的Chvatal-Gomory（CG）割和更一般的聚集割包装和覆盖整数规划（IP）的强度。聚合割的获得如下：给定一个IP公式，我们首先使用原始约束的聚合生成一个单一的隐含不等式，然后获得由这个单一不等式定义的集合的整数壳，最后使用描述整数壳的不等式作为切割平面。我们的第一个主要结果是表明，包装和覆盖的IP，CG和聚合闭包可以2-近似通过简单地生成各自的闭包为每个原始配方的约束，而不使用任何聚合。另一方面，我们使用计算实验表明，聚合削减可以任意强于削减从个人的限制一般的IP。证明上述结果的情况下，覆盖IP的界限需要发展一些新的结构性结果，这可能是独立的利益。最后，我们研究了基于k个不同的聚合不等式的割集的强度，即所谓的多行割集，并证明了具有大的完整性间隙的每个填充或覆盖IP也具有大的k-聚合闭包秩。特别地，该秩总是至少为完整性间隙的对数的量级。
英文摘要：
In this paper, we study the strength of Chvatal-Gomory (CG) cuts and more generally aggregation cuts for packing and covering integer programs (IPs). Aggregation cuts are obtained as follows: Given an IP formulation, we first generate a single implied inequality using aggregation of the original constraints, then obtain the integer hull of the set defined by this single inequality with variable bounds, and finally use the inequalities describing the integer hull as cutting-planes. Our first main result is to show that for packing and covering IPs, the CG and aggregation closures can be 2-approximated by simply generating the respective closures for each of the original formulation constraints, without using any aggregations. On the other hand, we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general IPs. The proof of the above stated results for the case of covering IPs with bounds require the development of some new structural results, which may be of independent interest. Finally, we examine the strength of cuts based on k different aggregation inequalities simultaneously, the so-called multi-row cuts, and show that every packing or covering IP with a large integrality gap also has a large k-aggregation closure rank. In particular, this rank is always at least of the order of the logarithm of the integrality gap.
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