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A Polynomial-Time Approximation Algorithm for Complete Interval Minors

Authors Romain Bourneuf , Julien Cocquet , Chaoliang Tang , Stéphan Thomassé

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LIPIcs.APPROX-RANDOM.2025.15.pdf
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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph theory
Keywords
  • Approximation algorithm
  • Ordered graphs
  • Interval minors
  • Delayed decompositions

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Abstract

As shown by Robertson and Seymour, deciding whether the complete graph K_t is a minor of an input graph G is a fixed parameter tractable problem when parameterized by t. From the approximation viewpoint, a substantial gap remains: there is no PTAS for finding the largest complete minor unless P = NP, whereas the best known result is a polytime O(√ n)-approximation algorithm by Alon, Lingas and Wahlén.
We investigate the complexity of finding K_t as interval minor in ordered graphs (i.e. graphs with a linear order on the vertices, in which intervals are contracted to form minors). Our main result is a polytime f(t)-approximation algorithm, where f is triply exponential in t but independent of n. The algorithm is based on delayed decompositions and shows that ordered graphs without a K_t interval minor can be constructed via a bounded number of three operations: closure under substitutions, edge union, and concatenation of a stable set. As a byproduct, graphs avoiding K_t as an interval minor have bounded chromatic number.

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Romain Bourneuf, Julien Cocquet, Chaoliang Tang, and Stéphan Thomassé. A Polynomial-Time Approximation Algorithm for Complete Interval Minors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 15:1-15:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.15

Author Details

Romain Bourneuf
  • Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400 Talence, France
Julien Cocquet
  • ENS de Lyon, Université Claude Bernard Lyon 1, CNRS, Inria, LIP UMR 5668, Lyon, France
Chaoliang Tang
  • Shanghai Center for Mathematical Science, Fudan University, Shanghai, China
  • ENS de Lyon, Université Claude Bernard Lyon 1, CNRS, Inria, LIP UMR 5668, Lyon, France
Stéphan Thomassé
  • ENS de Lyon, Université Claude Bernard Lyon 1, CNRS, Inria, LIP UMR 5668, Lyon, France

Funding

The authors are partially supported by the French National Research Agency under research grants ANR GODASse ANR-24-CE48-4377 and ANR Twin-width ANR-21-CE48-0014-01. The third author is also supported by Chinese Scholarship Council (CSC).

Acknowledgements

We thank Julien Duron for his help with the tedious calculations.

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