On Computational Aspects of Ordered Matching Problems
Michal Čertík, Andreas Emil Feldmann, Jaroslav Nešetřil, Paweł Rzążewski
TL;DR
The paper investigates computational and parameterized complexity aspects of problems on ordered matchings, including subgraph containment and homomorphisms. It establishes NP-completeness for SUB_<^M and Hom_<^M, while showing that when the image is restricted to an ordered matching, the Hom_< problem is fixed-parameter tractable (FPT) with respect to the size of the image, and this extends to colored variants and ordered H0-factors. It also proves that the core problem is polynomial-time solvable when the core is a matching, contrasting with NP-hardness in general CORE_<, and discusses colored extensions. The results delineate clear boundaries between intractable and tractable regimes and open questions about classifications for broader classes of ordered graphs.
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.