Subnetwork mining is an essential issue in the analysis of biological, social and communication networks. Recent applications require the simultaneous mining of several networks on the same or a similar vertex set. That is, one searches for subnetworks fulfilling different properties in each input network. We study the case that the input consists of a directed graph D and an undirected graph G on the same vertex set, and the sought pattern is a path P in D whose vertex set induces a connected subgraph of G. In this context, three concrete problems arise, depending on whether the existence of P is questioned or whether the length of P is to be optimized: in that case, one can search for a longest path or (maybe less intuitively) a shortest one. These problems have immediate applications in biological networks and predictable applications in social, information and communication networks. We study the classic and parameterized complexity of the problem, thus identifying polynomial and NP-complete cases, as well as fixed-parameter tractable and W[1]-hard cases. We also propose two enumeration algorithms that we evaluate on synthetic and biological data.
Abstract. Biological networks are commonly used to model molecular activity within the cell. Recent experimental studies have shown that the detection of conserved subnetworks across several networks, coming from different organisms, may allow the discovery of disease pathways and prediction of protein functions. There already exist automatic methods that allow to search for conserved subnetworks using networks alignment; unfortunately, these methods are limited to networks of same type, thus having the same graph representation. Towards overcoming this limitation, a unified framework for pairwise comparison and analysis of networks with different graph representations (in particular, a directed acyclic graph D and an undirected graph G over the same set of vertices) is presented in [4]. We consider here a related problem called k-DAGCC: given a directed graph D and an undirected graph G on the same set V of vertices, and an integer k, does there exist sets of vertices V1, V2, . . .is connected ? Two variants of k-DAGCC are of interest: (a) the Vis must form a partition of V , or (b) the Vis must form a cover of V . We study the computational complexity of both variants of k-DAGCC and, depending on the constraints imposed on the input, provide several polynomial-time algorithms, hardness and inapproximability results.
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