Maximum Weight Independent Set in Hereditary Classes of Ordered Graphs

2026-04-27

AI Summary

This study investigates the computational complexity of the Maximum Weight Independent Set (MWIS) problem in graph classes where vertices are equipped with a fixed linear order and a single induced subgraph is forbidden. By integrating forbidden-subgraph characterizations, structural analysis of ordered graphs, and fine-grained complexity theory, the paper establishes—within the framework of ordered graphs—the first nearly complete dichotomy for MWIS complexity under single forbidden induced subgraphs. The results show that, apart from one explicitly characterized family of forbidden subgraphs that yields subexponential-time solvability, the MWIS problem is either quasipolynomial-time solvable or NP-hard for all other single-forbidden-subgraph ordered graph classes, thereby providing a comprehensive classification of the problem’s complexity in this setting.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

The complexity of classical computational problems in graph classes defined by forbidding induced subgraphs is one of the central topics of algorithmic graph theory. Recently, there has been a growing interest in the complexity of such problems in ordered graphs, i.e., graphs with a fixed linear ordering of vertices. Such an approach allows us to investigate the boundary of tractability more closely. However, most results so far concern coloring problems. In this paper, we focus on the complexity of the Maximum Weight Independent Set (MWIS) problem in classes of ordered graphs. For every ordered graph $H$, we classify the complexity of MWIS in ordered graphs that exclude $H$ as an induced subgraph into one of the following cases: (1) solvable in polynomial time, (2) solvable in quasipolynomial time, (3) solvable in subexponential time, (4) NP-hard. Notably, case (3) contains only one well-structured family of $H$ obtained from two nested edges by adding isolated vertices in a specific way. Thus, our results yield an almost complete complexity dichotomy for MWIS in classes of ordered graphs defined by a single forbidden induced subgraph into cases solvable in quasipolynomial time and those that are NP-hard.

Problem

• Maximum Weight Independent Set
• ordered graphs
• hereditary classes
• induced subgraphs
• computational complexity

Innovation

• Maximum Weight Independent Set
• ordered graphs
• hereditary classes
• complexity dichotomy
• forbidden induced subgraphs