A Polynomial-Time Approximation Algorithm for Complete Interval Minors

2025-05-09

AI Summary

This paper addresses the problem of approximating the largest complete interval minor $K_t$ in an ordered graph. Prior work achieved only an $O(sqrt{n})$ approximation ratio, dependent on the input size $n$. We overcome this limitation by establishing the first structural characterization of $K_t$-interval-minor-free ordered graphs: such graphs can be constructed via three bounded operations—substitution closure, edge union, and stable-set gluing—implying they have bounded chromatic number. Leveraging this structural theory, we develop a novel polynomial-time $f(t)$-approximation algorithm, where $f(t)$ is a triple-exponential function of $t$. Our approach integrates delayed decomposition with interval-minor composition techniques. Crucially, the approximation ratio depends solely on $t$, eliminating any dependence on $n$. This yields the first $t$-only approximation for the problem, resolving a fundamental dependency barrier in ordered graph minor theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• Develops polynomial-time approximation for complete interval minors
• Addresses complexity of finding K_t in ordered graphs
• Provides algorithm based on delayed decompositions

Innovation

• Polytime f(t)-approximation algorithm for interval minors
• Uses delayed decompositions for ordered graphs
• Constructs graphs via three bounded operations