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**Published in:** LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

We study the NP-hard graph problem Collapsed k-Core where, given an undirected graph G and integers b, x, and k, we are asked to remove b vertices such that the k-core of remaining graph, that is, the (uniquely determined) largest induced subgraph with minimum degree k, has size at most x. Collapsed k-Core was introduced by Zhang et al. [AAAI 2017] and it is motivated by the study of engagement behavior of users in a social network and measuring the resilience of a network against user drop outs. Collapsed k-Core is a generalization of r-Degenerate Vertex Deletion (which is known to be NP-hard for all r >=0) where, given an undirected graph G and integers b and r, we are asked to remove b vertices such that the remaining graph is r-degenerate, that is, every its subgraph has minimum degree at most r.
We investigate the parameterized complexity of Collapsed k-Core with respect to the parameters b, x, and k, and several structural parameters of the input graph. We reveal a dichotomy in the computational complexity of Collapsed k-Core for k <=2 and k >= 3. For the latter case it is known that for all x >= 0 Collapsed k-Core is W[P]-hard when parameterized by b. We show that Collapsed k-Core is W[1]-hard when parameterized by b and in FPT when parameterized by (b+x) if k <=2. Furthermore, we show that Collapsed k-Core is in FPT when parameterized by the treewidth of the input graph and presumably does not admit a polynomial kernel when parameterized by the vertex cover number of the input graph.

Junjie Luo, Hendrik Molter, and Ondrej Suchý. A Parameterized Complexity View on Collapsing k-Cores. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{luo_et_al:LIPIcs.IPEC.2018.7, author = {Luo, Junjie and Molter, Hendrik and Such\'{y}, Ondrej}, title = {{A Parameterized Complexity View on Collapsing k-Cores}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {7:1--7:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.7}, URN = {urn:nbn:de:0030-drops-102088}, doi = {10.4230/LIPIcs.IPEC.2018.7}, annote = {Keywords: r-Degenerate Vertex Deletion, Feedback Vertex Set, Fixed-Parameter Tractability, Kernelization Lower Bounds, Graph Algorithms, Social Network Analysis} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

We introduce a dynamic version of the NP-hard Cluster Editing problem. The essential point here is to take into account dynamically evolving input graphs: Having a cluster graph (that is, a disjoint union of cliques) that represents a solution for a first input graph, can we cost-efficiently transform it into a "similar" cluster graph that is a solution for a second ("subsequent") input graph? This model is motivated by several application scenarios, including incremental clustering, the search for compromise clusterings, or also local search in graph-based data clustering. We thoroughly study six problem variants (edge editing, edge deletion, edge insertion; each combined with two distance measures between cluster graphs). We obtain both fixed-parameter tractability as well as parameterized hardness results, thus (except for two open questions) providing a fairly complete picture of the parameterized computational complexity landscape under the perhaps two most natural parameterizations: the distance of the new "similar" cluster graph to (i) the second input graph and to (ii) the input cluster graph.

Junjie Luo, Hendrik Molter, André Nichterlein, and Rolf Niedermeier. Parameterized Dynamic Cluster Editing. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{luo_et_al:LIPIcs.FSTTCS.2018.46, author = {Luo, Junjie and Molter, Hendrik and Nichterlein, Andr\'{e} and Niedermeier, Rolf}, title = {{Parameterized Dynamic Cluster Editing}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {46:1--46:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.46}, URN = {urn:nbn:de:0030-drops-99450}, doi = {10.4230/LIPIcs.FSTTCS.2018.46}, annote = {Keywords: graph-based data clustering, goal-oriented clustering, compromise clustering, NP-hard problems, fixed-parameter tractability, parameterized hardness} }

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