In this paper, we bring the techniques of the Laplacian paradigm to the congested clique, while further restricting ourselves to deterministic algorithms. In particular, we show how to solve a Laplacian system up to precision $\epsilon$ in $n^{o(1)}\log(1/\epsilon)$ rounds. We show how to leverage this result within existing interior point methods for solving flow problems. We obtain an $m^{3/7+o(1)}U^{1/7}$ round algorithm for maximum flow on a weighted directed graph with maximum weight $U$, and we obtain an $\tilde{O}(m^{3/7}(n^{0.158}+n^{o(1)}\text{poly}\log W))$ round algorithm for unit capacity minimum cost flow on a directed graph with maximum cost $W$. Hereto, we give a novel routine for computing Eulerian orientations in $O(\log n \log^* n)$ rounds, which we believe may be of separate interest.

翻译：在本文中，我们将拉普拉斯范式的技术引入到强拥塞团中，同时进一步将自身限制为确定性算法。具体而言，我们展示了如何在$n^{o(1)}\log(1/\epsilon)$轮内求解精度达到$\epsilon$的拉普拉斯系统。我们还展示了如何将这一结果应用于现有的内点法中，以解决流问题。对于最大权重为$U$的加权有向图上的最大流问题，我们得到了一种$m^{3/7+o(1)}U^{1/7}$轮的算法；对于最大成本为$W$的有向图上的单位容量最小成本流问题，我们得到了一种$\tilde{O}(m^{3/7}(n^{0.158}+n^{o(1)}\text{poly}\log W))$轮的算法。为此，我们提出了一种在$O(\log n \log^* n)$轮内计算欧拉定向的新颖例程，我们认为该例程可能具有独立的研究价值。