On Computational Aspects of Ordered Matching Problems

TL;DR

This paper investigates the computational complexity of ordered matching problems, establishing NP-completeness for some cases while identifying fixed-parameter tractability under certain conditions, and provides algorithms for related problems.

Contribution

It proves NP-completeness of ordered matching problems and shows fixed-parameter tractability when the image graph is an ordered matching, also offering algorithms for these problems.

Findings

Ordered matching subgraph problem is NP-complete.

Finding ordered homomorphisms is NP-complete.

Homomorphism problem is FPT when the image graph is an ordered matching.

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.