In the context of metabolic network analysis, Lacroix et al. 11 introduced the problem of finding occurrences of motifs in vertex-colored graphs, where a motif is a multiset of colors and an occurrence of a motif is a subset of connected vertices which are colored by all colors of the motif. We consider in this paper the above-mentioned problem in one of its natural optimization forms, referred hereafter as the Min-CC problem: Find an occurrence of a motif in a vertexcolored graph, called the target graph, that induces a minimum number of connected components. Our results can be summarized as follows. We prove the Min-CC problem to be APX–hard even in the extremal case where the motif is a set and the target graph is a path. We complement this result by giving a polynomial-time algorithm in case the motif is built upon a fixed number of colors and the target graph is a path. Also, extending recent research 8, we prove the Min-CC problem to be fixed-parameter tractable when parameterized by the size of the motif, and we give a faster algorithm in case the target graph is a tree. Furthermore, we prove the Min-CC problem for trees not to be approximable within ratio c log n for some constant c> 0, where n is the order of the target graph, and to be W[2]–hard when parameterized by the number of connected components in the occurrence of the motif. Finally, we give an exact efficient exponential-time algorithm for the Min-CC problem in case the target graph is a tree. 1

Abstract. We study the NP-complete Graph Motif problem: given a vertex-colored 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 |M | · |M | 2 · |E|). We extend our algorithm to list-colored 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 presumably simpler problems of finding (uncolored) biconnected or bridge-connected 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 graphs that are paths. 1

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 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 logδ n unless NP ⊆ DTIME(2 poly 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 efficient exponential-time algorithms for the problem. 1

Complex networks from domains like Biology or Sociology are present in many e-Science 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 pseudo-code 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.

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 NP-hard 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 vertex-colored 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 above-mentioned problem in two of its natural optimization forms, referred hereafter as the Min-CC and the Maximum Motif problems. The Min CC problem seeks for an occurrence of a motif M in a vertex-colored graph G that induces a minimum