clustering challenges in biological networks - s. butenko (world, 2009)

[...]... Ernst-Abbe-Platz 2, D-07743 Jena, Germany hueffner@minet.uni-jena.de Rolf Niedermeier Institut f¨ r Informatik, Friedrich-Schiller-Universit¨ t Jena u a Ernst-Abbe-Platz 2, D-07743 Jena, Germany niedermr@minet.uni-jena.de Web: http://theinf1.informatik.uni-jena.de/ Sebastian Wernicke† Institut f¨ r Informatik, Friedrich-Schiller-Universit¨ t Jena u a Ernst-Abbe-Platz 2, D-07743 Jena, Germany wernicke@minet.uni-jena.de... Volume - 9in x 6in PART 1 Surveys of Selected Topics 1 clustering November 11, 2008 16:23 World Scientific Review Volume - 9in x 6in This page intentionally left blank 2 clustering November 11, 2008 16:23 World Scientific Review Volume - 9in x 6in Chapter 1 Fixed-Parameter Algorithms for Graph-Modeled Data Clustering Falk H¨ ffner∗ u Institut f¨ r Informatik, Friedrich-Schiller-Universit¨ t Jena u a Ernst-Abbe-Platz... the solution set and removing it and its incident edges from the instance clustering November 11, 2008 16:23 World Scientific Review Volume - 9in x 6in Fixed-Parameter Algorithms for Graph-Modeled Data Clustering clustering 9 1.2.1.2 The Kernelization Concept Abstractly speaking, what have we done in the previous section? After applying a number of rules in polynomial time to an instance of V ERTEX C OVER,... originated from studies of this single problem (e.g., see Guo et al [40]) b V ERTEX clustering November 11, 2008 16:23 World Scientific Review Volume - 9in x 6in Fixed-Parameter Algorithms for Graph-Modeled Data Clustering clustering 7 rameter k—and ask whether there is an algorithm that confines the combinatorial explosion that is involved in solving the problem to the parameter Definition 1.1 An instance... E) and a nonnegative clustering November 11, 2008 16:23 World Scientific Review Volume - 9in x 6in Fixed-Parameter Algorithms for Graph-Modeled Data Clustering clustering 13 I H G Fig 1.3 A graph G with a crown I ∪H The thick edges constitute a maximum matching of size |H| in the bipartite graph that is induced by the edges between I and H integer k TASK : Find a k-vertex clique in G It is also a common... complicated branching with lots of case distinctions that might not be of practical value In the following, we describe the key observations and fundamental ideas behind the improved search trees for C LUSTER E DITING As a remark, there also clustering November 11, 2008 16:23 World Scientific Review Volume - 9in x 6in Fixed-Parameter Algorithms for Graph-Modeled Data Clustering clustering 17 exist size-o(3k )... solution that is obtained More precisely, a size-n instance of a fixed-parameter tractable problem can be solved in f (k) · p(n) time, where f is a function solely depending on k, and p(n) is a polynomial in n clustering November 11, 2008 16:23 World Scientific Review Volume - 9in x 6in Fixed-Parameter Algorithms for Graph-Modeled Data Clustering clustering 5 The purpose of this survey is twofold: First,... “optimum” value of k is most efficient to employ in practice clustering November 11, 2008 16:23 World Scientific Review Volume - 9in x 6in clustering Fixed-Parameter Algorithms for Graph-Modeled Data Clustering 11 k k−1 k−2 k−1 k−2 k−2 k−2 Fig 1.2 Simple search tree for finding a vertex cover of size at most k in a given graph The size of the tree is upper-bounded by O(2k ) one of these two vertices... connection to the fixed-parameter tractable V ERTEX C OVER problem that we used as our running example to introduce fixed-parameter techniques: If an n-vertex graph contains a size-k clique, then its complement graphg contains a size-(n−k) vertex cover and vice versa This can be made use of when seeking after or enumerating cliques 1.3.1.1 Finding Maximum Cardinality Cliques The catch when solving C LIQUE for... of vertices linear in k g That is, the graph that contains exactly those edges that are not contained in the original graph November 11, 2008 14 16:23 World Scientific Review Volume - 9in x 6in H¨ ffner, Niedermeier & Wernicke u A crown in a graph consists of an independent set I (that is, no two vertices in I are connected by an edge) and a set H containing all vertices adjacent to I In order for . AggregateAssociations 282 15.3. SubjectiveAssociations 286 15.4. Case-SpecificAssociations 287 15.5. RelationsbetweenMeasures 288 15.6. ClusteringDrugs 290 15.7. InterpretingtheClusters 298 15.8. Summary. outliers. Clustering algorithms have been successfully used to analyze the data sets arising in many important applications of diverse origin, including biology. In fact, applications of clustering. 36]. Practical experiments that we discuss in the case studies suggest that the pre- sented fixed-parameter algorithms are capable of solving many real-world in- stances in reasonable time. In particular,