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Minimum Cost Nowhere-Zero Flows and Cut-Balanced Orientations

Authors Karthekeyan Chandrasekaran , Siyue Liu , R. Ravi

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Routing and network design problems
Keywords
  • Nowhere-zero Flows
  • Cut-balanced orientations
  • Bicriteria approximation algorithms

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Abstract

Flows and colorings are disparate concepts in graph algorithms - the former is tractable while the latter is intractable. Tutte [Tutte, 1954; Tutte, 1966] introduced the concept of nowhere-zero flows to unify these two concepts. Jaeger [Jaeger, 1976] showed that nowhere-zero flows are equivalent to cut-balanced orientations. Motivated by connections between nowhere-zero flows, cut-balanced orientations, Nash-Williams' well-balanced orientations, and postman problems, we study optimization versions of nowhere-zero flows and cut-balanced orientations. Given a bidirected graph with asymmetric costs on two orientations of each edge, we study the min cost nowhere-zero k-flow problem and min cost k-cut-balanced orientation problem. We show that both problems are NP-hard to approximate within any finite factor. Given the strong inapproximability result, we design bicriteria approximations for both problems: we obtain a (6,6)-approximation to the min cost nowhere-zero k-flow and a (k,6)-approximation to the min cost k-cut-balanced orientation. For the case of symmetric costs (where the costs of both orientations are the same for every edge), we show that the nowhere-zero k-flow problem remains NP-hard and admits a 3-approximation.

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Karthekeyan Chandrasekaran, Siyue Liu, and R. Ravi. Minimum Cost Nowhere-Zero Flows and Cut-Balanced Orientations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 46:1-46:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.46

Author Details

Karthekeyan Chandrasekaran
  • Grainger College of Engineering, University of Illinois, Urbana-Champaign, IL, USA
Siyue Liu
  • Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA
R. Ravi
  • Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA

Funding

Karthekeyan Chandrasekaran is supported in part by NSF grant CCF-2402667. Siyue Liu and R. Ravi are supported in part by the Air Force Office of Scientific Research under award number FA9550-23-1-0031.

Acknowledgements

The first two authors would like to thank Emléktábla workshop in July 2024 and ICERM reunion workshop on Discrete Optimization: Mathematics, Algorithms, and Computation in August 2024, where this work was initiated. We would like to thank Kristóf Bérczi and Gérard Cornuéjols for initial discussions.

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