论文标题
更快的小树宽SDP求解器
A Faster Small Treewidth SDP Solver
论文作者
论文摘要
半决赛编程是优化和理论计算机科学的基本工具。它已被广泛用作解决许多问题的黑框，例如嵌入，复杂性，学习和差异。 半宽度设置的半自然设置。小树宽设置下的最佳先前的SDP求解器是Zhang-lavaei '18，该''''采用$ n^{1.5}τ^{6.5} $ time。在这项工作中，我们展示了如何使用$ n \ times n $变量，$ m $约束和$τ$ treewidth在$nτ^{2Ω+0.5} $ time中求解半程编程，其中$ω<2.373 $表示矩阵乘法的指数。我们给出了第一个在此设置下以变量数量的线性及时运行的SDP求解器。 此外，我们将用Tau Treewidth求解线性编程的运行时间从$Nτ^2 $（Dong-Lee-Ye '21）到$Nτ^{（ω+1）/2} $。
Semidefinite programming is a fundamental tool in optimization and theoretical computer science. It has been extensively used as a black-box for solving many problems, such as embedding, complexity, learning, and discrepancy. One natural setting of semidefinite programming is the small treewidth setting. The best previous SDP solver under small treewidth setting is due to Zhang-Lavaei '18, which takes $n^{1.5} τ^{6.5}$ time. In this work, we show how to solve a semidefinite programming with $n \times n$ variables, $m$ constraints and $τ$ treewidth in $n τ^{2ω+0.5}$ time, where $ω< 2.373$ denotes the exponent of matrix multiplication. We give the first SDP solver that runs in time in linear in number of variables under this setting. In addition, we improve the running time that solves a linear programming with tau treewidth from $n τ^2$ (Dong-Lee-Ye '21) to $n τ^{(ω+1)/2}$.
