稠密图中的更快高精度多商品流算法
Faster high-accuracy multicommodity flow in dense graphs
- Carnegie Mellon University（卡内基梅隆大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出一种求解最小成本k商品流问题的快速算法，通过构造低秩辅助线性规划实现至少1/k的进展，在稠密有向图上运行时间为多项式因子乘以n^2.5+m√n，不依赖快速矩阵乘法。
AI中文摘要：
我们给出了一种求解最小成本$k$商品流问题的快速算法。基本思想是构造一个低秩的辅助线性规划，其最小值至多为原问题最小值的$1/k$倍（因此，求解该辅助线性规划将在原问题上取得至少$1/k$的进展）。低秩线性规划可以使用黑盒技术快速求解。因此，我们的算法在具有$n$个顶点和$m$条边的有向图上运行时间为$\tilde{O}(\text{poly}(k)(n^{2.5}+m\sqrt{n}))$。我们不依赖任何快速矩阵乘法。
英文摘要：
We give a fast algorithm for solving min-cost $k$-commodity flow. The basic idea is to construct an auxiliary linear program that has low rank and whose minimum value is at most $1/k$ times the minimum value of the original problem (thus, solving this auxiliary linear program will make at least $1/k$ fraction of progress in the original problem). Low-rank linear programs can be solved quickly using black-box techniques. Thus, our algorithm runs in time $\tilde{O}(\operatorname{poly}(k)(n^{2.5}+m\sqrt{n}))$ on a directed graph with $n$ vertices and $m$ edges. We do not rely on any fast matrix multiplication.
