图划分问题的常数因子Lasserre积分间隙
Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems
摘要：
将图的顶点划分为两个大致相等的部分，同时最小化穿过切割的边的数量是一个基本问题（称为平衡分离器），在许多设置中都会出现。对于这个问题，以及诸如均匀稀疏切割问题的变体，其中目标是最小化切割的相对侧上由边缘连接的对的分数，在已知的近似算法和非近似性结果之间存在很大的差距。虽然没有常数因子近似算法是已知的，甚至APX硬度也是未知的。在这项工作中，我们证明了平衡分离器和均匀稀疏割，半定程序从Lasserre层次结构（这是最强大的松弛研究文献中）有一个完整的差距有界远离$1$，即使是$Ω(n)$水平的层次结构。这补充了Guruswami和Sinop（2011）最近的算法结果，后者使用Lasserre层次结构为这些问题提供了一个近似方案（运行时间取决于图的谱）。在此过程中，我们观察到，简化了提升“多项式约束”（如平衡分隔符中的全局平衡约束）到Lasserre层次结构的更高级别的任务。
英文摘要：
Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such as the Uniform Sparsest Cut problem where the goal is to minimize the fraction of pairs on opposite sides of the cut that are connected by an edge, there are large gaps between the known approximation algorithms and non-approximability results. While no constant factor approximation algorithms are known, even APX-hardness is not known either. In this work we prove that for balanced separator and uniform sparsest cut, semidefinite programs from the Lasserre hierarchy (which are the most powerful relaxations studied in the literature) have an integrality gap bounded away from $1$, even for $Ω(n)$ levels of the hierarchy. This complements recent algorithmic results in Guruswami and Sinop (2011) which used the Lasserre hierarchy to give an approximation scheme for these problems (with runtime depending on the spectrum of the graph). Along the way, we make an observation that simplifies the task of lifting "polynomial constraints" (such as the global balance constraint in balanced separator) to higher levels of the Lasserre hierarchy.
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