目标为PSD的二次规划的Lasserre族、高阶特征值和逼近格式
Lasserre Hierarchy, Higher Eigenvalues, and Approximation Schemes for Quadratic Integer Programming with PSD Objectives
摘要：
本文提出了一种求解具有半正定目标函数和全局线性约束的二次规划问题的近似算法。这个框架包括著名的图问题，如最小图二等分，边扩展，均匀稀疏割，小集扩展，以及唯一游戏问题。这些问题是臭名昭著的存在巨大的差距之间的已知算法的结果和NP-难的结果。我们的算法是基于四舍五入半定程序从拉塞尔层次，和分析使用的边界低秩近似的矩阵Frobenius范数使用列的矩阵。对于上述所有图问题，我们给出了一个具有逼近比$\frac{1+ε}{\min\{1,λ_r\}}$的在时间上运行的算法$n^{O(r/ε^2)}$，其中$λ_r$是规范化图Laplacian $\mathcal{L}$的第$r$个最小特征值.在图的二分和小集扩张的情况下，割中的顶点数在规定界限的低阶项内。我们的结果意味着$(1+O(ε))$因子在时间上的近似$n^{O(r^\ast/ε^2)}$其中$r^\ast$是小于$1-ε$的$\mathcal{L}$的特征值的数目。对于唯一对策，我们给出了一个因子$(1+\frac{2+ε}{λ_r})$近似，用于最小化$n^{O(r/ε)}$时间内未满足约束的数量。这改进了在扩展器上求解唯一博弈的早期边界，也表明Lasserre SDP足够强大，可以解决基本SDP的著名完整性缺口实例。我们也给出了一个算法的独立集的图，表现良好的拉普拉斯算子没有太多的特征值大于$1+o(1)$。
英文摘要：
We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These problems are notorious for the existence of huge gaps between the known algorithmic results and NP-hardness results. Our algorithm is based on rounding semidefinite programs from the Lasserre hierarchy, and the analysis uses bounds for low-rank approximations of a matrix in Frobenius norm using columns of the matrix. For all the above graph problems, we give an algorithm running in time $n^{O(r/ε^2)}$ with approximation ratio $\frac{1+ε}{\min\{1,λ_r\}}$, where $λ_r$ is the $r$'th smallest eigenvalue of the normalized graph Laplacian $\mathcal{L}$. In the case of graph bisection and small set expansion, the number of vertices in the cut is within lower-order terms of the stipulated bound. Our results imply $(1+O(ε))$ factor approximation in time $n^{O(r^\ast/ε^2)}$ where $r^\ast$ is the number of eigenvalues of $\mathcal{L}$ smaller than $1-ε$. For Unique Games, we give a factor $(1+\frac{2+ε}{λ_r})$ approximation for minimizing the number of unsatisfied constraints in $n^{O(r/ε)}$ time. This improves an earlier bound for solving Unique Games on expanders, and also shows that Lasserre SDPs are powerful enough to solve well-known integrality gap instances for the basic SDP. We also give an algorithm for independent sets in graphs that performs well when the Laplacian does not have too many eigenvalues bigger than $1+o(1)$.
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