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On tractable cases of Target Set Selection

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Abstract

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.

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Notes

  1. Informally speaking, W[1]-hardness means that there is no hope for fixed-parameter tractability with respect to the corresponding parameter; we refer to Sect. 2 for the formal definitions.

  2. Given an undirected graph G = (V, E) and a positive integer h, the Dominating Set problem asks whether there is a vertex subset \(V^{\prime}\subseteq V\) of size at most h such that for every \(v\in V\) it holds that \(v \in V^{\prime}\) or v is adjacent to a vertex in \(V^{\prime}. \)

  3. For example, this is plausible for networks of teams (say in sports or work teams) where vertices are persons and there is an edge between two vertices if the intersection of their team membership durations is sufficiently large.

  4. Given an undirected graph G = (V, E) and a positive integer h, the Vertex Cover problem asks whether there is a vertex subset \(V^{\prime}\subseteq V\) of size at most h such that each edge in E has at least one endpoint in \(V^{\prime}.\)

  5. The dataset and a corresponding documentation are available online (http://dblp.uni-trier.de/xml/).

  6. These three networks are available in the Pajek Dataset of Vladimir Batagelj and Andrej Mrvar (2006) (http://vlado.fmf.uni-lj.si/pub/networks/data/).

  7. It is well-known that a parameterized problem is fixed-parameter tractable if and only if it has a kernelization.

  8. The cluster edge deletion number ξ is lower bounded by the cluster editing number \(\zeta, \) i.e., \(\xi \geq \zeta. \)

  9. A bridge of a connected graph G is an edge whose deletion disconnects G.

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Authors and Affiliations

  1. Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Berlin, Germany

    André Nichterlein, Rolf Niedermeier, Johannes Uhlmann & Mathias Weller

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  1. André Nichterlein
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  2. Rolf Niedermeier
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  3. Johannes Uhlmann
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  4. Mathias Weller
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Correspondence to André Nichterlein.

Additional information

This work is supported by the DFG, research projects PABI, NI 369/7, and DARE, NI 369/11. An extended abstract appeared in the Proceedings of the 21st International Symposium on Algorithms and Computation (ISAAC ’10), Part I, volume 6506 of LNCS, pages 378–389, Springer 2010.

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Nichterlein, A., Niedermeier, R., Uhlmann, J. et al. On tractable cases of Target Set Selection. Soc. Netw. Anal. Min. 3, 233–256 (2013). https://doi.org/10.1007/s13278-012-0067-7

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