A Simple Efficient Interior Point Method for Min-Cost Flow

Abstract

We present a novel simpler method for the min-cost flow problem and prove that its expected running time is bounded by $\tilde{O}(m^{3/2})$. This matches the best known bounds, which have previously been achieved only by far more complex algorithms or by algorithms for special cases. Our contribution contains three algorithmic parts that are interesting in their own right: (1) We provide a linear time construction of an equivalent auxiliary network and interior primal and dual points, i.e. flows, node potentials and slacks, with potential $P_0=\tilde{O}(\sqrt{m})$. (2) We present a potential reduction algorithm that transforms initial solutions of potential $P_0$ to ones with duality gap below $1$ in $\tilde{O}(P_0\cdot \text{ CEF }(n,m,\epsilon ))$ time, where $\epsilon ^{-1}=O(m^2)$ and $\text{ CEF }(n,m,\epsilon )$ denotes the running time of any algorithm that computes an $\varepsilon$-approximate electrical flow. (3) We show that, taking solutions with duality gap less than $1$ as input, one can compute optimal integral node potentials in $O(m+n\log n)$ time with our novel crossover procedure. Altogether, using a variant of a state-of-the-art $\varepsilon$-electrical flow solver, we obtain a new simple algorithm for the min-cost flow problem running in $\tilde{O}(m^{3/2})$.