On the parameterized complexity of multiple-interval graph problems

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multiple-interval graphs, often allowing for more robustness in the modeling of the specific applicat...

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Vydané v:Theoretical computer science Ročník 410; číslo 1; s. 53 - 61
Hlavní autori: Fellows, Michael R., Hermelin, Danny, Rosamond, Frances, Vialette, Stéphane
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Oxford Elsevier B.V 01.01.2009
Elsevier
Predmet:
Applied sciences
Calculus of variations and optimal control
Clique
Computer Science
Computer science; control theory; systems
Data Structures and Algorithms
Dominating set
Exact sciences and technology
Independent set
Information retrieval. Graph
Mathematical analysis
Mathematics
Miscellaneous
Multicolored clique
Multiple intervals
Numerical analysis
Numerical analysis. Scientific computation
Numerical methods in mathematical programming, optimization and calculus of variations
Parameterized complexity
Sciences and techniques of general use
Theoretical computing
W-hardness
Multiple intervals
Dominating set
W-hardness
Multicolored clique
Independent set
Clique
Parameterized complexity
Interval graph
Vertex
Computer theory
Optimization method
Unit
Modeling
Optimization
Complexity
Robustness
Application
ISSN:0304-3975, 1879-2294
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Popis
Shrnutí:Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multiple-interval graphs, often allowing for more robustness in the modeling of the specific application. With this motivation in mind, a recent systematic study of optimization problems in multiple-interval graphs was initiated. In this sequel, we study multiple-interval graph problems from the perspective of parameterized complexity. The problems under consideration are k - Independent Set, k - Dominating Set, and k - Clique, which are all known to be W[1]-hard for general graphs, and NP-complete for multiple-interval graphs. We prove that k - Clique is in FPT, while k - Independent Set and k - Dominating Set are both W[1]-hard. We also prove that k - Independent Dominating Set, a hybrid of the two above problems, is also W[1]-hard. Our hardness results hold even when each vertex is associated with at most two intervals, and all intervals have unit length. Furthermore, as an interesting byproduct of our hardness results, we develop a useful technique for showing W[1]-hardness via a reduction from the k - Multicolored Clique problem, a variant of k - Clique. We believe this technique has interest in its own right, as it should help in simplifying W[1]-hardness results which are notoriously hard to construct and technically tedious.
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2008.09.065