CPM 2009: Combinatorial Pattern Matching pp 221-235

# Maximum Motif Problem in Vertex-Colored Graphs

• Riccardo Dondi
• Guillaume Fertin
• Stéphane Vialette
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5577)

## Abstract

Searching for motifs in graphs has become a crucial problem in the analysis of biological networks. In this context, different graph motif problems have been considered [13,7,5]. Pursuing a line of research pioneered by Lacroix et al. [13], we introduce in this paper a new graph motif problem: given a vertex colored graph G and a motif $$\mathcal{M}$$, where a motif is a multiset of colors, find a maximum cardinality submotif $$\mathcal{M}' \subseteq \mathcal{M}$$ that occurs as a connected motif in G. We prove that the problem is APX-hard even in the case where the target graph is a tree of maximum degree 3, the motif is actually a set and each color occurs at most twice in the tree. Next, we strengthen this result by proving that the problem is not approximable within factor $$2^{\rm {log^{\delta} n}}$$, for any constant δ< 1, unless NPDTIMEclass(2POLY log n). We complement these results by presenting two fixed-parameter algorithms for the problem, where the parameter is the size of the solution. Finally, we give exact fast exponential-time algorithms for the problem.

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## Authors and Affiliations

• Riccardo Dondi
• 1
• Guillaume Fertin
• 2
• Stéphane Vialette
• 3
1. 1.Dipartimento di Scienze dei Linguaggi, della Comunicazione e degli Studi CulturaliUniversità degli Studi di BergamoBergamoItaly
2. 2.Laboratoire d’Informatique de Nantes-Atlantique (LINA), UMR CNRS 6241Université de NantesNantes Cedex 3France
3. 3.IGM-LabInfo, CNRS UMR 8049Université Paris-EstMarne-la-ValléeFrance