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19
Parameterized algorithms and hardness results for some graph motif problems
 Combinatorial Pattern Matching, volume 5029 of Lecture Notes in Computer Science
, 2008
"... Abstract. We study the NPcomplete Graph Motif problem: given a vertexcolored graph G = (V, E) and a multiset M of colors, does there exist an S ⊆ V such that G[S] is connected and carries exactly (also with respect to multiplicity) the colors in M ? We present an improved randomized algorithm for ..."
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Cited by 17 (2 self)
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Abstract. We study the NPcomplete Graph Motif problem: given a vertexcolored graph G = (V, E) and a multiset M of colors, does there exist an S ⊆ V such that G[S] is connected and carries exactly (also with respect to multiplicity) the colors in M ? We present an improved randomized algorithm for Graph Motif with running time O(4.32 . We extend our algorithm to listcolored graph vertices and the case where the motif G[S] needs not be connected. By way of contrast, we show that extending the request for motif connectedness to the somewhat "more robust" motif demands of biconnectedness or bridgeconnectedness leads to W[1]complete problems. Actually, we show that the even simpler problems of finding biconnected or bridgeconnected subgraphs are W[1]complete with respect to the subgraph size. Answering an open question from the literature, we further show that the parameter number of connected motif components leads to W[1]hardness even when restricted to the very special case of graphs that are paths.
Strategies for Network Motifs Discovery
 FIFTH IEEE INTERNATIONAL CONFERENCE ON ESCIENCE
, 2009
"... Complex networks from domains like Biology or Sociology are present in many eScience data sets. Dealing with networks can often form a workflow bottleneck as several related algorithms are computationally hard. One example is detecting characteristic patterns or “network motifs” – a problem involv ..."
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Cited by 16 (10 self)
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Complex networks from domains like Biology or Sociology are present in many eScience data sets. Dealing with networks can often form a workflow bottleneck as several related algorithms are computationally hard. One example is detecting characteristic patterns or “network motifs” – a problem involving subgraph mining and graph isomorphism. This paper provides a review and runtime comparison of current motif detection algorithms in the field. We present the strategies and the corresponding algorithms in pseudocode yielding a framework for comparison. We categorize the algorithms outlining the main differences and advantages of each strategy. We finally implement all strategies in a common platform to allow a fair and objective efficiency comparison using a set of benchmark networks. We hope to inform the choice of strategy and critically discuss future improvements in motif detection.
Upper and Lower Bounds for Finding Connected Motifs in VertexColored Graphs
, 2010
"... We study the problem of finding occurrences of motifs in vertexcolored graphs, where a motif is a multiset of colors, and an occurrence of a motif is a subset of connected vertices whose multiset of colors equals the motif. This problem is a natural graphtheoretic pattern matching variant where we ..."
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Cited by 12 (3 self)
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We study the problem of finding occurrences of motifs in vertexcolored graphs, where a motif is a multiset of colors, and an occurrence of a motif is a subset of connected vertices whose multiset of colors equals the motif. This problem is a natural graphtheoretic pattern matching variant where we are not interested in the actual structure of the occurrence of the pattern, we only require it to preserve the very basic topological requirement of connectedness. We give two positive results and three negative results that together give an extensive picture of tractable and intractable instances of the problem.
Weak pattern matching in colored graphs: Minimizing the number of connected components
, 2007
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Finding and Counting VertexColored Subtrees
 INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE (MFCS), BRNO: CZECH REPUBLIC
, 2010
"... The problems studied in this article originate from the Graph Motif problem introduced by Lacroix et al. [17] in the context of biological networks. The problem is to decide if a vertexcolored graph has a connected subgraph whose colors equal a given multiset of colors M. Using an algebraic framewo ..."
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Cited by 9 (2 self)
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The problems studied in this article originate from the Graph Motif problem introduced by Lacroix et al. [17] in the context of biological networks. The problem is to decide if a vertexcolored graph has a connected subgraph whose colors equal a given multiset of colors M. Using an algebraic framework recently introduced by Koutis et al. [15,16], we obtain new FPT algorithms for Graph Motif and variants, with improved running times. We also obtain results on the counting versions of this problem, showing that the counting problem is FPT if M is a set, but becomes #W[1]hard if M is a multiset with two colors.
Maximum motif problem in vertexcolored graphs
 In Proc. 20th CPM, volume 5577 of LNCS
, 2009
"... 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 [12, 6, 4]. Pursuing a line of research pioneered by Lacroix et al. [12], we introduce in this paper a new graph motif pr ..."
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Cited by 7 (1 self)
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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 [12, 6, 4]. Pursuing a line of research pioneered by Lacroix et al. [12], we introduce in this paper a new graph motif problem: given a vertex colored graph G and a motif M, where a motif is a multiset of colors, find a maximum cardinality submotif M ′ ⊆ M that occurs as a connected motif in G. We prove that the problem is APXhard 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 logδ n unless NP ⊆ DTIME(2 poly log n). We complement these results by presenting two fixedparameter algorithms for the problem, where the parameter is the size of the solution. Finally, we give exact efficient exponentialtime algorithms for the problem. 1
Probably Optimal Graph Motifs
, 2013
"... We show an O ∗ (2 k)time polynomial space algorithm for the ksized Graph Motif problem. We also introduce a new optimization variant of the problem, called Closest Graph Motif and solve it within the same time bound. The Closest Graph Motif problem encompasses several previously studied optimizati ..."
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Cited by 6 (2 self)
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We show an O ∗ (2 k)time polynomial space algorithm for the ksized Graph Motif problem. We also introduce a new optimization variant of the problem, called Closest Graph Motif and solve it within the same time bound. The Closest Graph Motif problem encompasses several previously studied optimization variants, like Maximum Graph Motif, MinSubstitute, and MinAdd. Moreover, we provide a piece of evidence that our result might be essentially tight: the existence of an O ∗ ((2−ɛ) k)time algorithm for the Graph Motif problem implies an O((2−ɛ ′ ) n)time algorithm for Set Cover.
Steijger T: Annotating fragmentation patterns
 In Proc. of Workshop on Algorithms in Bioinformatics (WABI 2009), volume 5724 of Lect Notes Comput Sci
"... Abstract. Mass spectrometry is one of the key technologies in metabolomics for the identification and quantification of molecules in small concentrations. For identification, these molecules are fragmented by, e.g., tandem mass spectrometry, and masses and abundances of the resulting fragments are m ..."
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Abstract. Mass spectrometry is one of the key technologies in metabolomics for the identification and quantification of molecules in small concentrations. For identification, these molecules are fragmented by, e.g., tandem mass spectrometry, and masses and abundances of the resulting fragments are measured. Recently, methods for de novo interpretation of tandem mass spectra and the automated inference of fragmentation patterns have been developed. If the correct structural formula is known, then peaks in the fragmentation pattern can be annotated by substructures of the underlying compound. To determine the structure of these fragments manually is tedious and timeconsuming. Hence, there is a need for automated identification of the generated fragments. In this work, we consider the problem of annotating fragmentation patterns. Our input are fragmentation trees, representing tandem mass spectra where each peak has been assigned a molecular formula, and fragmentation dependencies are known. Given a fixed structural formula and any fragment molecular formula, we search for all structural fragments that satisfy elemental multiplicities. Ultimately, we search for a fragmentation pattern annotation with minimum total cleavage costs. We discuss several algorithmic approaches for this problem, including a randomized and a tree decompositionbased algorithm. We find that even though the problem of identifying structural fragments is NPhard, instances based on molecular structures can be efficiently solved with a classical branchandbound algorithm. 1
GraMoFoNe: a Cytoscape plugin for querying motifs without topology in ProteinProtein Interactions networks
 In 2nd International Conference on Bioinformatics and Computational Biology (BICoB’10
, 2010
"... During the last decade, data on ProteinProtein Interactions (PPI) has increased in a huge manner. Searching for motifs in PPI Network has thus became a crucial problem to interpret this data. A large part of the literature is devoted to the query of motifs with a given topology. However, the biolog ..."
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Cited by 5 (2 self)
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During the last decade, data on ProteinProtein Interactions (PPI) has increased in a huge manner. Searching for motifs in PPI Network has thus became a crucial problem to interpret this data. A large part of the literature is devoted to the query of motifs with a given topology. However, the biological data are, by now, so noisy (missing and erroneous information) that the topology of a motif can be unrelevant. Consequently, Lacroix et al. [19] defined a new problem, called GRAPH MOTIF, which consists in searching a multiset of colors in a vertexcolored graph. In this article, we present GraMoFoNe, a plugin to Cytoscape based on a Linear PseudoBoolean optimization solver which handles GRAPH MOTIF and some of its extensions. 1.
Complexity Issues in VertexColored Graph Pattern Matching
, 2010
"... Searching for motifs in graphs has become a crucial problem in the analysis of biological networks. In the context of metabolic network analysis, Lacroix et al [V. Lacroix, C.G. Fernandes and M.F. Sagot, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 3 (2006), no. 4, 360368] int ..."
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Cited by 4 (1 self)
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Searching for motifs in graphs has become a crucial problem in the analysis of biological networks. In the context of metabolic network analysis, Lacroix et al [V. Lacroix, C.G. Fernandes and M.F. Sagot, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 3 (2006), no. 4, 360368] introduced the NPhard general problem of finding occurrences of motifs in vertexcolored graphs, where a motif M is a multiset of colors and an occurrence of M in a vertexcolored graph G, called the target graph, is a subset of vertices that induces a connected graph and the multiset of colors induces by this subset is exactly the motif. Pursuing the line of research pioneered by Lacroix et al. and aiming at dealing with approximate solutions, we consider in this paper the abovementioned problem in two of its natural optimization forms, referred hereafter as the MinCC and the Maximum Motif problems. The Min CC problem seeks for an occurrence of a motif M in a vertexcolored graph G that induces a minimum