We consider a new semidefinite programming relaxation for directed edge expansion, which is obtained by adding triangle inequalities to the reweighted eigenvalue formulation. Applying the matrix multiplicative weight update method to this relaxation, we derive almost linear-time algorithms to achieve $O(\sqrt{\log{n}})$-approximation and Cheeger-type guarantee for directed edge expansion, as well as an improved cut-matching game for directed graphs. This provides a primal-dual flow-based framework to obtain the best known algorithms for directed graph partitioning. The same approach also works for vertex expansion and for hypergraphs, providing a simple and unified approach to achieve the best known results for different expansion problems and different algorithmic techniques.

翻译：我们考虑了一种新的用于有向边扩展的半定规划松弛，该松弛通过将三角不等式加入重加权特征值公式而获得。将此矩阵乘性权重更新方法应用于该松弛，我们推导出近乎线性时间的算法，以实现有向边扩展的$O(\sqrt{\log{n}})$-近似和Cheeger型保证，以及一种改进的有向图割匹配博弈。这提供了一个原始-对偶的基于流的框架，用以获得已知最佳的有向图划分算法。同样的方法也适用于顶点扩展和超图，为不同扩展问题和不同算法技术获得已知最佳结果提供了一种简单统一的途径。