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arXiv · 2505.05997

A Polynomial-Time Approximation Algorithm for Complete Interval Minors

Romain Bourneuf·Julien Cocquet·Chaoliang Tang·Stéphan Thomassé
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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, the gap to fill is quite large, as there is no PTAS for finding the largest complete minor unless $P = NP$, whereas a polytime $O(\sqrt n)$-approximation algorithm was given by Alon, Lingas and Wahl\'en. 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, Stéphan Thomassé. 2025-05-09. A Polynomial-Time Approximation Algorithm for Complete Interval Minors. https://arxiv.org/abs/2505.05997

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