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

Authors Sally Dong, Guanghao Ye

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Author Details

Sally Dong
  • University of Washington, Seattle, WA, USA
Guanghao Ye
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Ye, Guanghao: Supported by NSF awards CCF-1955217 and DMS-2022448.

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Sally Dong and Guanghao Ye. Faster Min-Cost Flow and Approximate Tree Decomposition on Bounded Treewidth Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.49

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 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Network optimization
Keywords
  • Min-cost flow
  • tree decomposition
  • interior point method
  • bounded treewidth graphs

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