arXiv 1207.43722012-07cs.DS

局部舍入算法的快速SDP层次求解器

Faster SDP hierarchy solvers for local rounding algorithms

Venkatesan Guruswami, Ali Kemal Sinop

备注：: 30 pages, 8 figures

摘要：

基于不同层次的线性/半定规划的凸松弛最近已被用于设计各种优化问题的近似算法。这些算法的近似保证随着层级中的{ \em轮} $r$的数量而提高，尽管求解（或甚至写下其解）$r$的第一级程序的复杂度随着$n^{Ω(r)}$而增长，其中$n$是输入大小。在这项工作中，我们观察到，许多这些算法是基于{ \em本地}舍入程序，只使用一小部分的SDP解决方案（大小$n^{O(1)} 2^{O(r)}$，而不是$n^{Ω(r)}$）。我们给出了一个算法，以找到所需的部分在时间多项式的大小。实现这一目标的挑战在于，解决方案的所需部分不是先验固定的，而是取决于解决方案的其他部分，有时是以复杂的迭代方式。我们的求解器导致$n^{O(1)} 2^{O(r)}$时间算法，以获得相同的保证，在许多情况下，早期的$n^{O(r)}$时间算法的基础上$r$轮的Lasserre层次。特别地，基于$O(\log n)$轮的保证可以在多项式时间内实现。我们开发和描述我们的算法在一个相当一般的抽象框架。在我们的工作中的主要技术工具，这可能是独立的利益，凸优化，是一个有效的椭圆算法为基础的分离预言凸计划，可以输出{ \em证书的不可行性与限制支持}。这是在一个递归的方式来找到一个序列的一致性点嵌套凸体，“傻瓜”本地舍入算法。

英文摘要：

Convex relaxations based on different hierarchies of linear/semi-definite programs have been used recently to devise approximation algorithms for various optimization problems. The approximation guarantee of these algorithms improves with the number of {\em rounds} $r$ in the hierarchy, though the complexity of solving (or even writing down the solution for) the $r$'th level program grows as $n^{Ω(r)}$ where $n$ is the input size. In this work, we observe that many of these algorithms are based on {\em local} rounding procedures that only use a small part of the SDP solution (of size $n^{O(1)} 2^{O(r)}$ instead of $n^{Ω(r)}$). We give an algorithm to find the requisite portion in time polynomial in its size. The challenge in achieving this is that the required portion of the solution is not fixed a priori but depends on other parts of the solution, sometimes in a complicated iterative manner. Our solver leads to $n^{O(1)} 2^{O(r)}$ time algorithms to obtain the same guarantees in many cases as the earlier $n^{O(r)}$ time algorithms based on $r$ rounds of the Lasserre hierarchy. In particular, guarantees based on $O(\log n)$ rounds can be realized in polynomial time. We develop and describe our algorithm in a fairly general abstract framework. The main technical tool in our work, which might be of independent interest in convex optimization, is an efficient ellipsoid algorithm based separation oracle for convex programs that can output a {\em certificate of infeasibility with restricted support}. This is used in a recursive manner to find a sequence of consistent points in nested convex bodies that "fools" local rounding algorithms.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文