Johannes Uhlmann

On tractable cases of Target Set Selection

We study the NP-hard Target Set Selection (TSS) problem occurring in social network analysis. Roughly speaking, given a graph where each vertex is associated with a threshold, in TSS the task is to select a minimum-size “target set” such that all vertices of the graph get activated. Activation is a dynamic process. First, only the vertices in the target set are active. Then, a vertex becomes active if the number of its active neighbors exceeds its threshold, and so on. TSS models the spread of information, infections, and influence in networks. Complementing results on its polynomial-time approximability and extending results for its restriction to trees and bounded treewidth graphs, we classify the influence of the parameters “diameter”, “cluster editing number”, “vertex cover number”, and “feedback edge set number” of the underlying graph on the problem’s computational complexity, revealing both tractable and intractable cases. For instance, even for diameter-two split graphs TSS remains W[2]-hard with respect to the parameter “size of the target set”. TSS can be efficiently solved on graphs with small feedback edge set number and also turns out to be fixed-parameter tractable when parameterized by the vertex cover number. Both results contrast known parameterized intractability results for the parameter “treewidth”. While these tractability results are relevant for sparse networks, we also show efficient fixed-parameter algorithms for the parameter “cluster editing number”, yielding tractability for certain dense networks.

A More Relaxed Model for Graph-Based Data Clustering:

We introduce the

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-Plex Cluster Editing

-Plex Cluster Editing problem as a generalization of the well-studied Cluster Editing problem; both are NP-hard and both are motivated by graph-based data clustering. Instead of transforming a given graph by a minimum number of edge modifications into a disjoint union of cliques (this is Cluster Editing), the task in the case of

On Bounded-Degree Vertex Deletion parameterized by treewidth

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Complexity and exact algorithms for vertex multicut in interval and bounded treewidth graphs

-Plex Cluster Editing is to transform a graph into a cluster graph consisting of a disjoint union of so-called

A More Relaxed Model for Graph-Based Data Clustering: s-Plex Editing

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Partitioning into colorful components by minimum edge deletions

-plexes. Herein, an

Two-Layer planarization parameterized by feedback edge set

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On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

-plex is a vertex set