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Algorithms 2015, 8(1), 60-81; doi:10.3390/a8010060

The Parameterized Complexity of the Rainbow Subgraph Problem

Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Ernst-Reuter-Platz 7, D-10587 Berlin, Germany
This paper is an extended version of our paper published in the Proceedings of the 40th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2014), Nouan-le-Fuzelier, France, 25–27 June 2014.
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Academic Editor: Jesper Jansson
Received: 22 September 2014 / Revised: 13 February 2015 / Accepted: 16 February 2015 / Published: 27 February 2015
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Abstract

The NP-hard RAINBOW SUBGRAPH problem, motivated from bioinformatics, is to find in an edge-colored graph a subgraph that contains each edge color exactly once and has at most \(k\) vertices. We examine the parameterized complexity of RAINBOW SUBGRAPH for paths, trees, and general graphs. We show that RAINBOW SUBGRAPH is W[1]-hard with respect to the parameter \(k\) and also with respect to the dual parameter \(\ell:=n-k\) where \(n\) is the number of vertices. Hence, we examine parameter combinations and show, for example, a polynomial-size problem kernel for the combined parameter \(\ell\) and ``maximum number of colors incident with any vertex''. Additionally, we show APX-hardness even if the input graph is a properly edge-colored path in which every color occurs at most twice. View Full-Text
Keywords: APX-hardness; multivariate complexity analysis; fixed-parameter tractability; parameterized hardness; problem kernel; haplotyping APX-hardness; multivariate complexity analysis; fixed-parameter tractability; parameterized hardness; problem kernel; haplotyping
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Hüffner, F.; Komusiewicz, C.; Niedermeier, R.; Rötzschke, M. The Parameterized Complexity of the Rainbow Subgraph Problem. Algorithms 2015, 8, 60-81.

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