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Back in the Saddle: Toward Parallel Approximate Minimum-Cost Flow

Authors Rasmus Kyng , Aurelio L. Sulser

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LIPIcs.ICALP.2026.136.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Convex optimization
Keywords
  • Approximate Min-Cost-Flow
  • Parallel
  • Area-Convexity
  • Expanders

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Abstract

We present the first polylog-depth, nearly-linear-work parallel algorithm that achieves a (1+ε)-bicriteria approximation guarantee for undirected minimum-cost flow on expanders. Fix an undirected graph G = (V,E) with unit capacities, unit lengths, and conductance ϕ. For any feasible demand vector d and any ε ∈ (0,1) we compute, in Õ(|E|/(εϕ)) work and Õ(1/(εϕ)) depth, a flow f that routes d exactly while satisfying ‖f‖_∞ ≤ 1+ε and ‖f‖_1 ≤ (1+ε)min_{Bg = d, ‖g‖_∞ ≤ 1}}‖g‖_1.
This bicriteria guarantee simultaneously controls congestion and total cost, strengthens the previously studied notion of throughput error, and matches the best known ε-dependence for parallel maximum flow/transshipment on general graphs.
Our main contribution is a new saddle-point optimization method for mixed 𝓁_∞-𝓁_1 optimization. Concretely, we (i) formulate a two-term regression capturing minimum-cost flow as a saddle-point problem that couples 𝓁_∞ and 𝓁_1 terms, (ii) construct a small-magnitude area-convex regularizer tailored to the resulting primal–dual domain (building on Sherman’s area-convexity framework [Sherman, 2017]), and (iii) implement efficient δ-approximate maximization/minimization oracles so that Sherman’s extragradient iteration yields low iteration-count convergence. 
Beyond the concrete expander result, our mixed 𝓁_∞-𝓁_1 optimization toolkit appears broadly applicable and suggests a promising route toward Õ(m/ε) work and Õ(1/ε) depth algorithms for approximate undirected minimum-cost flow on general graphs.

Cite As Get BibTex

Rasmus Kyng and Aurelio L. Sulser. Back in the Saddle: Toward Parallel Approximate Minimum-Cost Flow. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 136:1-136:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.136

Author Details

Rasmus Kyng
  • ETH Zurich, Switzerland
Aurelio L. Sulser
  • ETH Zurich, Switzerland

Funding

The research leading to these results has received funding from the starting grant "A New Paradigm for Flow and Cut Algorithms" (no. TMSGI2_218022) of the Swiss National Science Foundation.

References

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