A Combinatorial Cut-Based Algorithm for Solving Laplacian Linear Systems
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Over the last two decades, a significant line of work in theoretical algorithms has been progress in solving linear systems of the form 𝐋𝐩 = 𝐛, where 𝐋 is the Laplacian matrix of a weighted graph with weights w(i,j)>0 on the edges. The solution 𝐩 of the linear system can be interpreted as the potentials of an electrical flow. Kelner, Orrechia, Sidford, and Zhu <cit.> give a combinatorial, near-linear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law. Here we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law. We prove that this dual algorithm also runs in a near-linear number of iterations. Each iteration requires updating all potentials on one side of a fundamental cut of a spanning tree by a fixed amount. If this update step can be performed in polylogarithmic time, we can also obtain a near-linear time algorithm to solve 𝐋𝐩 = 𝐛. However, if we abstract this update step as a natural data structure problem, we show that we can use the data structure to solve a problem that has been conjectured to be difficult for dynamic algorithms, the online vector-matrix-vector problem <cit.>. The conjecture implies that the data structure does not have an O(n^1-ϵ) time algorithm for any ϵ > 0. Thus our dual algorithm cannot be near-linear time algorithm for solving 𝐋𝐩 = 𝐛 unless we are able to take advantage of the structure of the particular update steps that our algorithm uses.
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