A Polynomial-Time Approximation Algorithm for Complete Interval Minors

TL;DR

This paper presents a polynomial-time approximation algorithm for detecting complete minors as interval minors in ordered graphs, advancing understanding of graph minor problems with practical algorithmic implications.

Contribution

It introduces a triply exponential approximation algorithm for complete interval minors in ordered graphs, using delayed decompositions and specific graph operations.

Findings

Algorithm runs in polynomial time with approximation factor f(t).

Ordered graphs avoiding K_t as an interval minor have bounded chromatic number.

Provides a structural characterization of graphs without K_t interval minors.

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.