The multicolored graph realization problem
| Autor: | Josep Díaz, Öznur Yaşar Diner, Maria Serna, Oriol Serra |
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| Přispěvatelé: | Universitat Politècnica de Catalunya. Departament de Ciències de la Computació, Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. ALBCOM - Algorísmia, Bioinformàtica, Complexitat i Mètodes Formals, Universitat Politècnica de Catalunya. GAPCOMB - Geometric, Algebraic and Probabilistic Combinatorics |
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: |
FOS: Computer and information sciences
Discrete Mathematics (cs.DM) Grafs Teoria de Applied Mathematics Generalized combinatorial problems Computational Complexity (cs.CC) Convex bipartite graphs Graph theory Computer Science - Computational Complexity Parameterized complexity Informàtica::Informàtica teòrica [Àrees temàtiques de la UPC] Discrete Mathematics and Combinatorics MathematicsofComputing_DISCRETEMATHEMATICS Computer Science - Discrete Mathematics Multicolored realization problem |
| Popis: | We introduce the Multicolored Graph Realization problem (MGRP). The input to the problem is a colored graph $(G,\varphi)$, i.e., a graph together with a coloring on its vertices. We can associate to each colored graph a cluster graph ($G_\varphi)$ in which, after collapsing to a node all vertices with the same color, we remove multiple edges and self-loops. A set of vertices $S$ is multicolored when $S$ has exactly one vertex from each color class. The problem is to decide whether there is a multicolored set $S$ such that, after identifying each vertex in $S$ with its color class, $G[S]$ coincides with $G_\varphi$. The MGR problem is related to the class of generalized network problems, most of which are NP-hard. For example the generalized MST problem. MGRP is a generalization of the Multicolored Clique Problem, which is known to be W[1]-hard when parameterized by the number of colors. Thus MGRP remains W[1]-hard, when parameterized by the size of the cluster graph and when parameterized by any graph parameter on $G_\varphi$, among those for treewidth. We look to instances of the problem in which both the number of color classes and the treewidth of $G_\varphi$ are unbounded. We show that MGRP is NP-complete when $G_\varphi$ is either chordal, biconvex bipartite, complete bipartite or a 2-dimensional grid. Our hardness results follows from suitable reductions from the 1-in-3 monotone SAT problem. Our reductions show that the problem remains hard even when the maximum number of vertices in a color class is 3. In the case of the grid, the hardness holds also graphs with bounded degree. We complement those results by showing combined parameterizations under which the MGR problem became tractable. Comment: 23 pages, 9 figures |
| Databáze: | OpenAIRE |
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