A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow

Jan van den Brand; Li Chen; Rasmus Kyng; Yang P. Liu; Richard Peng; Maximilian Probst Gutenberg; Sushant Sachdeva; Aaron Sidford

arXiv · 2309.16629

A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow

Jan van den Brand·Li Chen·Rasmus Kyng·Yang P. Liu·Richard Peng·Maximilian Probst Gutenberg·Sushant Sachdeva·Aaron Sidford

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Abstract

We give a deterministic $m^{1+o(1)}$ time algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with $m$ edges and polynomially bounded integral demands, costs, and capacities. As a consequence, we obtain the first running time improvement for deterministic algorithms that compute maximum-flow in graphs with polynomial bounded capacities since the work of Goldberg-Rao [J.ACM '98]. Our algorithm builds on the framework of Chen-Kyng-Liu-Peng-Gutenberg-Sachdeva [FOCS '22] that computes an optimal flow by computing a sequence of $m^{1+o(1)}$-approximate undirected minimum-ratio cycles. We develop a deterministic dynamic graph data-structure to compute such a sequence of minimum-ratio cycles in an amortized $m^{o(1)}$ time per edge update. Our key technical contributions are deterministic analogues of the vertex sparsification and edge sparsification components of the data-structure from Chen et al. For the vertex sparsification component, we give a method to avoid the randomness in Chen et al. which involved sampling random trees to recurse on. For the edge sparsification component, we design a deterministic algorithm that maintains an embedding of a dynamic graph into a sparse spanner. We also show how our dynamic spanner can be applied to give a deterministic data structure that maintains a fully dynamic low-stretch spanning tree on graphs with polynomially bounded edge lengths, with subpolynomial average stretch and subpolynomial amortized time per edge update.

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Jan van den Brand, Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, Sushant Sachdeva, Aaron Sidford. 2023-09-28. A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow. https://arxiv.org/abs/2309.16629

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