eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-09-23
49:1
49:14
10.4230/LIPIcs.ESA.2024.49
article
Faster Min-Cost Flow and Approximate Tree Decomposition on Bounded Treewidth Graphs
Dong, Sally
1
Ye, Guanghao
2
https://orcid.org/0000-0002-1197-0649
University of Washington, Seattle, WA, USA
Massachusetts Institute of Technology, Cambridge, MA, USA
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 Õ(m√{τ} + S) time. This improves upon the previous fastest algorithm in this setting achieved by the bounded-treewidth linear program solver of [Gu and Song, 2022; Dong et al., 2024], which runs in Õ(m τ^{(ω+1)/2}) time, where ω ≈ 2.37 is the matrix multiplication exponent. Our approach leverages recent advances in structured linear program solvers and robust interior point methods (IPM). In general graphs where treewidth is trivially bounded by n, the algorithm runs in Õ(m √ n) time, which is the best-known result without using the Lee-Sidford barrier or 𝓁₁ IPM, demonstrating the surprising power of robust interior point methods.
As a corollary, we obtain a Õ(tw³ ⋅ m) time algorithm to compute a tree decomposition of width O(tw⋅ log(n)), given a graph with m edges.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol308-esa2024/LIPIcs.ESA.2024.49/LIPIcs.ESA.2024.49.pdf
Min-cost flow
tree decomposition
interior point method
bounded treewidth graphs