# Faster Min-Cost Flow on Bounded Treewidth Graphs

We present a O(m√(τ)+nτ) time algorithm for finding a minimum-cost flow in graphs with n vertices and m edges, given a tree decomposition of width τ and polynomially bounded integer costs and capacities. This improves upon the current best algorithms for general linear programs bounded by treewidth which run in O(m τ^(ω+1)/2) time by [Dong-Lee-Ye,21] and [Gu-Song,22], where ω≈ 2.37 is the matrix multiplication exponent. Our approach leverages recent advances in structured linear program solvers and robust interior point methods. As a corollary, for any graph G with n vertices, m edges, and treewidth τ, we obtain a O(τ^3 · m) time algorithm to compute a tree decomposition of G with width O(τ·log n).

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