通过稳定分布求解划分拟阵约束下的行列式最大化
Determinant maximization subject to a partition matroid constraint via stable distributions
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对划分拟阵约束下的行列式最大化问题，提出基于稳定分布的多项式时间近似算法，将鞍点松弛转化为多线性松弛，达到与最优值估计匹配的近似保证。
AI中文摘要：
给定向量 $v_i \in {\mathbb R}^d$，我们考虑在划分拟阵中选择一个独立集 $I$ 以最大化行列式 $\det (\sum_{i \in I} v_i v_i^T)$ 的问题。我们的主要结果是一个多项式时间近似算法，该算法找到一个解，其值为 $\det ( \sum_{i \in I} v_{i} v_{i}^T) \geq e^{-O(d)} OPT$，其中 $OPT = \max_{I^*} \det ( \sum_{i \in I^*} v_{i} v_{i}^T)$。对于秩 $m \leq d$ 的划分拟阵，我们给出了类似的结果，用于近似所选向量张成的 $m$ 维体积，近似因子为 $e^{O(m)}$。这与先前已知的估计最优值但不找到相应解的算法相匹配，仅在指数上的常数有所差异。与这些估计算法类似，我们的算法基于 Nikolov 和 Singh 提出的鞍点松弛。一个新的要素是基于 $1/2$-稳定分布的随机化变换，该变换将鞍点松弛转化为更方便的多线性松弛。
英文摘要：
Given vectors $v_i \in {\mathbb R}^d$, we consider the problem of choosing a set $I$ independent in a partition matroid in order to maximize the determinant $\det (\sum_{i \in I} v_i v_i^T)$. Our main result is a polynomial-time approximation algorithm that finds a solution of value $det ( \sum_{i \in I} v_{i} v_{i}^T) \geq e^{-O(d)} OPT$, where $OPT = \max_{I^*} det ( \sum_{i \in I^*} v_{i} v_{i}^T)$. For partition matroids of rank $m \leq d$, we give a similar result for approximating the $m$-dimensional volume spanned by the chosen vectors, within a factor of $e^{O(m)}$. This matches earlier known algorithms that estimate the optimal value but do not find the corresponding solution, up to a constant in the exponent. Similar to these estimation algorithms, our algorithm is based on the saddle-point relaxation proposed by Nikolov and Singh. A new ingredient is a randomized transformation based on $1/2$-stable distributions, which converts the saddle-point relaxation into a more convenient multilinear relaxation.
