Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems whose coefficient matrix is the Laplacian matrix of a weighted graph. The solution of the linear system can be interpreted as the potentials of an electrical flow. Kelner, Orrechia, Sidford, and Zhu (STOC 2013) give a combinatorial, near-linear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law by updating flows around cycles (cycle toggling). In this paper, we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law by cut toggling: each iteration updates all potentials on one side of a fundamental cut of a spanning tree by the same amount. We prove that this dual algorithm also runs in a near-linear number of iterations. We show, however, that if we abstract cut toggling as a natural data structure problem, this problem can be reduced to the online vector-matrix-vector problem, which has been conjectured to be difficult for dynamic algorithms by Henzinger, Krinninger, Nanongkai, and Saranurak (STOC 2015). The conjecture implies that a straightforward implementation of the cut-toggling algorithm requires essentially linear time per iteration. To circumvent the lower bound, we batch update steps, and perform them simultaneously instead of sequentially. An appropriate choice of batching leads to an $\tilde{O}(m^{1.5})$ time cut-toggling algorithm for solving Laplacian systems. Furthermore, if we sparsify the graph and call our algorithm recursively on the Laplacian system implied by batching and sparsifying, the running time can be reduced to $O(m^{1+\epsilon})$ for any $\epsilon > 0$. Thus, the dual cut-toggling algorithm can achieve (almost) the same running time as its primal cycle-toggling counterpart.

翻译：在过去的二十年中，理论算法领域的一项重大研究进展致力于求解系数矩阵为加权图拉普拉斯矩阵的线性系统。该线性系统的解可解释为电网络中的电势分布。Kelner、Orrechia、Sidford 和 Zhu（STOC 2013）提出了一种组合型、近线性时间的算法，该算法维持基尔霍夫电流定律，并通过围绕回路更新流量（回路切换）逐步强制执行基尔霍夫电压定律。本文考虑该算法的对偶版本：它维持基尔霍夫电压定律，并通过割切换逐步强制执行基尔霍夫电流定律——每次迭代对生成树的一个基本割一侧的所有电势施加相同的增量。我们证明该对偶算法同样在近线性次迭代内收敛。然而，若将割切换抽象为一个自然的数据结构问题，则该问题可归约为在线向量-矩阵-向量积问题，而Henzinger、Krinninger、Nanongkai 和 Saranurak（STOC 2015）曾猜想该问题对动态算法而言是困难的。该猜想意味着割切换算法的直接实现每次迭代本质上需要线性时间。为突破此下界，我们采用批处理更新步骤，将顺序更新改为同步执行。通过合理选择批处理规模，可得到时间复杂度为$\tilde{O}(m^{1.5})$的割切换算法以求解拉普拉斯系统。此外，若对图进行稀疏化，并在由批处理与稀疏化导出的拉普拉斯系统上递归调用我们的算法，则运行时间可降至对任意$\epsilon > 0$均为$O(m^{1+\epsilon})$。因此，对偶割切换算法能达到与其原始回路切换算法（几乎）相同的运行性能。