A Potential Reduction Inspired Algorithm for Exact Max Flow in Almost O(m^4/3) Time
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We present an algorithm for computing s-t maximum flows in directed graphs in O(m^4/3+o(1)U^1/3) time. Our algorithm is inspired by potential reduction interior point methods for linear programming. Instead of using scaled gradient/Newton steps of a potential function, we take the step which maximizes the decrease in the potential value subject to advancing a certain amount on the central path, which can be efficiently computed. This allows us to trace the central path with our progress depending only ℓ_∞ norm bounds on the congestion vector (as opposed to the ℓ_4 norm required by previous works) and runs in O(√(m)) iterations. To improve the number of iterations by establishing tighter bounds on the ℓ_∞ norm, we then consider the weighted central path framework of Madry <cit.> and Liu-Sidford <cit.>. Instead of changing weights to maximize energy, we consider finding weights which maximize the maximum decrease in potential value. Finally, similar to finding weights which maximize energy as done in <cit.> this problem can be solved by the iterative refinement framework for smoothed ℓ_2-ℓ_p norm flow problems <cit.> completing our algorithm. We believe our potential reduction based viewpoint provides a versatile framework which may lead to faster algorithms for max flow.
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