Almost-Linear Time Algorithms for Incremental Graphs: Cycle Detection, SCCs, s-t Shortest Path, and Minimum-Cost Flow

We give the first almost-linear time algorithms for several problems in incremental graphs including cycle detection, strongly connected component maintenance, s-t shortest path, maximum flow, and minimum-cost flow. To solve these problems, we give a deterministic data structure that returns a mo(1)-approximate minimum-ratio cycle in fully dynamic graphs in amortized mo(1) time per update. Combining this with the interior point method framework of Brand-Liu-Sidford (STOC 2023) gives the first almost-linear time algorithm for deciding the first update in an incremental graph after which the cost of the minimum-cost flow attains value at most some given threshold F. By rather direct reductions to minimum-cost flow, we are then able to solve the problems in incremental graphs mentioned above.