Faster Min-Cost Flow and Approximate Tree Decomposition on Bounded Treewidth Graphs
Sally Dong, Guanghao Ye
TL;DR
This work delivers a faster min-cost flow algorithm on graphs with bounded treewidth by marrying a robust interior point method with separator-tree based data structures that maintain sparse Schur complements and block-wise IPM updates. It extends the planarFlow framework to $\tau$-separable graphs, achieving a runtime of $\widetilde{O}(m\sqrt{\tau}\log M + S)$ given a tree decomposition of width $\tau$ and size $S$, and provides a corollary for approximating tree decompositions in $\widetilde{O}(\mathrm{tw}^3\cdot m)$ time. The approach hinges on recursive Schur-complement maintenance via a $\tau$-separator tree, block-coordinate updates, and periodic restarts to bound amortized costs, illustrating the power of robust IPMs combined with separator-based structures for structured LPs and min-cost flow problems.
Abstract
We present an algorithm for min-cost flow in graphs with $n$ vertices and $m$ edges, given a tree decomposition of width $τ$ and size $S$, and polynomially bounded, integral edge capacities and costs, running in $\widetilde{O}(m\sqrtτ + S)$ 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 $\widetilde{O}(m τ^{(ω+1)/2})$ time, where $ω\approx 2.37$ 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.