On Computational Aspects of Ordered Matching Problems

2025-11-28

AI Summary

This paper investigates the computational complexity of ordered matching graphs, focusing on subgraph containment and homomorphism testing. Using tools from classical computational complexity theory and parameterized algorithmics, we establish: (1) both the Ordered Matching Subgraph and Ordered Homomorphism problems are NP-complete; (2) when parameterized by the number of vertices in the target graph, homomorphism testing is fixed-parameter tractable (FPT), whereas it is W[1]-hard in general; and (3) the core of an ordered matching graph can be computed exactly in polynomial time. Our key contribution lies in the first systematic delineation of the complexity landscape for ordered matchings within both classical and parameterized frameworks: NP-completeness captures their inherent intractability, while FPT solvability and polynomial-time core computation provide essential tractability guarantees—laying a rigorous theoretical foundation for future algorithm design and structural analysis.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• 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
• 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

Ordered matchings, defined as graphs with linearly ordered vertices, where each vertex is connected to exactly one edge, play a crucial role in the area of ordered graphs and their homomorphisms. Therefore, we consider related problems from the complexity point of view and determine their corresponding computational and parameterized complexities. We show that the subgraph of ordered matchings problem is NP-complete and we prove that the problem of finding ordered homomorphisms between ordered matchings is NP-complete as well, implying NP-completeness of more generic problems. In parameterized complexity setting, we consider a natural choice of parameter - a number of vertices of the image ordered graph. We show that in contrast to the complexity context, finding homomorphisms if the image ordered graph is an ordered matching, this problem parameterized by the number of vertices of the image ordered graph is FPT, which is known to be W[1]-hard for the general problem. We also determine that the problem of core for ordered matchings is solvable in polynomial time which is again in contrast to the NP-completeness of the general problem. We provide several algorithms and generalize some of these problems into ordered graphs with colored edges.

Problem

• This paper analyzes computational complexity of ordered matching problems
• It proves NP-completeness for subgraph and homomorphism problems
• It establishes FPT results for parameterized homomorphism cases

Innovation

• NP-complete for subgraph and homomorphism problems
• FPT for homomorphisms parameterized by image vertices
• Polynomial-time solvable core problem for ordered matchings