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Exact and approximation algorithms for densest ksubgraph
, 2012
"... The densest ksubgraph problem is a generalization of the maximum clique problem, in which we are given a graph G and a positive integer k, and we search among the subsets of k vertices of G one inducing a maximum number of edges. In this paper, we present algorithms for finding exact solutions of d ..."
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The densest ksubgraph problem is a generalization of the maximum clique problem, in which we are given a graph G and a positive integer k, and we search among the subsets of k vertices of G one inducing a maximum number of edges. In this paper, we present algorithms for finding exact solutions of densest ksubgraph improving the standard exponential time complexity of O ∗ (2 n) and using polynomial space. Two FPT algorithms are also proposed; the first considers as parameter the treewidth of the input graph and uses exponential space, while the second is parameterized by the size of the minimum vertex cover and uses polynomial space. Finally, we propose several approximation algorithms running in moderately exponential or parameterized time.
Dynamical systems for discovering protein complexes and functional modules from biological networks
 IEEE/ACM Trans Comput Biol Bioinform
"... Abstract—Recent advances in high throughput experiments and annotations via published literature have provided a wealth of interaction maps of several biomolecular networks, including metabolic, proteinprotein, and proteinDNA interaction networks. The architecture of these molecular networks revea ..."
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Abstract—Recent advances in high throughput experiments and annotations via published literature have provided a wealth of interaction maps of several biomolecular networks, including metabolic, proteinprotein, and proteinDNA interaction networks. The architecture of these molecular networks reveals important principles of cellular organization and molecular functions. Analyzing such networks, i.e., discovering dense regions in the network, is an important way to identify protein complexes and functional modules. This task has been formulated as the problem of finding heavy subgraphs, the Heaviest kSubgraph Problem (kHSP), which itself is NPhard. However, any method based on the kHSP requires the parameter k and an exact solution of kHSP may still end up as a “spurious ” heavy subgraph, thus reducing its practicability in analyzing large scale biological networks. We proposed a new formulation, called the rankHSP, and two dynamical systems to approximate its results. In addition, a novel metric, called the Standard deviation and Mean Ratio (SMR), is proposed for use in “spurious ” heavy subgraphs to automate the discovery by setting a fixed threshold. Empirical results on both the simulated graphs and biological networks have demonstrated the efficiency and effectiveness of our proposal. Index Terms—Graph algorithms, neural nets, evolutionary computing, bioinformatics databases. Ç
The maximum node clustering problem
"... Dans cet article, nous introduisons un problème de graphes, appelé Maximum Node Clustering (MNC). Nous prouvons que ce problème est fortementNPcomplet et montrons qu’il peut être approché en temps polynomial avec un rapport arbitrairement proche de 2. Pour le cas particulier où le graphe est un ar ..."
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Dans cet article, nous introduisons un problème de graphes, appelé Maximum Node Clustering (MNC). Nous prouvons que ce problème est fortementNPcomplet et montrons qu’il peut être approché en temps polynomial avec un rapport arbitrairement proche de 2. Pour le cas particulier où le graphe est un arbre, nous prouvons que le problème est NPcomplet au sens faible puisqu’il généralise le problème du sacàdos et qu’on peut le résoudre en temps pseudopolynomial par une approche de programmation dynamique. Nous présentons également un FPTAS pour le cas des arbres. Motsclefs: Maximum Node Clustering, Sacàdos, Complexité, Approximation In this paper we introduce a graph problem, called Maximum Node Clustering (MNC). We prove that the problem is strongly NPcomplete and show that it can be approximated in polynomial time within a ratio arbitrarily close to 2. For the special case where the graph is a tree, we prove that the problem is weakly NPcomplete as it generalizes the 0/1 Knapsack problem and is solvable in pseudopolynomial time by a dynamic programming approach. For this latter case an FPTAS is also presented.
Discrete Applied Mathematics ( ) – Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics
"... journal homepage: www.elsevier.com/locate/dam A polyhedral study of the maximum edge subgraph problem ..."
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journal homepage: www.elsevier.com/locate/dam A polyhedral study of the maximum edge subgraph problem