Faster Min-Cost Flow and Approximate Tree Decomposition on Bounded Treewidth Graphs

TL;DR

This paper introduces a faster algorithm for min-cost flow on graphs with bounded treewidth, leveraging advanced linear programming techniques, and also provides an improved method for approximate tree decomposition.

Contribution

It presents a novel algorithm that improves min-cost flow computation time on bounded treewidth graphs and offers a new approach for approximate tree decomposition.

Findings

Achieves $ ilde{O}(m\sqrt{ au} + S)$ time for min-cost flow with tree decomposition.

Runs in $ ilde{O}(m \sqrt{n})$ time for general graphs, surpassing previous methods.

Provides a $ ilde{O}( ext{tw}^3 imes m)$ time algorithm for approximate tree decomposition.

Abstract

We present an algorithm for min-cost flow in graphs with $n$ vertices and $m$ edges, given a tree decomposition of width $\tau$ and size $S$, and polynomially bounded, integral edge capacities and costs, running in $\widetilde{O}(m\sqrt{\tau} + S)$ time. This improves upon the previous fastest algorithm in this setting achieved by the bounded-treewidth linear program solver by [Dong-Lee-Ye,21] and [Gu-Song,22], which runs in $\widetilde{O}(m \tau^{(\omega+1)/2})$ time, where $\omega \approx 2.37$ is the matrix multiplication exponent. Our approach leverages recent advances in structured linear program solvers and robust interior point methods (IPM). For general graphs where treewidth is trivially bounded by $n$, the algorithm runs in $\widetilde{O}(m \sqrt n)$ time, which is the best-known result without using the Lee-Sidford barrier or $\ell_1$ IPM, demonstrating the surprising power of robust interior point methods. As a corollary, we obtain a $\widetilde{O}(\operatorname{tw}^3 \cdot m)$ time algorithm to compute a tree decomposition of width $O(\operatorname{tw}\cdot \log(n))$, given a graph with $m$ edges.