{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11473","name":"A Parameterized Complexity View on Collapsing k-Cores","abstract":"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. \nWe 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.","keywords":["r-Degenerate Vertex Deletion","Feedback Vertex Set","Fixed-Parameter Tractability","Kernelization Lower Bounds","Graph Algorithms","Social Network Analysis"],"author":[{"@type":"Person","name":"Luo, Junjie","givenName":"Junjie","familyName":"Luo","affiliation":"Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Berlin, Germany, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China , School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China","funding":"Supported by CAS-DAAD Joint Fellowship Program for Doctoral Students of UCAS."},{"@type":"Person","name":"Molter, Hendrik","givenName":"Hendrik","familyName":"Molter","affiliation":"Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Berlin, Germany","funding":"Supported by the DFG, project MATE (NI 369\/17)."},{"@type":"Person","name":"Such\u00fd, Ondrej","givenName":"Ondrej","familyName":"Such\u00fd","affiliation":"Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic","funding":"Supported by grant 17-20065S of the Czech Science Foundation."}],"position":7,"pageStart":"7:1","pageEnd":"7:14","dateCreated":"2019-02-05","datePublished":"2019-02-05","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Luo, Junjie","givenName":"Junjie","familyName":"Luo","affiliation":"Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Berlin, Germany, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China , School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China","funding":"Supported by CAS-DAAD Joint Fellowship Program for Doctoral Students of UCAS."},{"@type":"Person","name":"Molter, Hendrik","givenName":"Hendrik","familyName":"Molter","affiliation":"Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Berlin, Germany","funding":"Supported by the DFG, project MATE (NI 369\/17)."},{"@type":"Person","name":"Such\u00fd, Ondrej","givenName":"Ondrej","familyName":"Such\u00fd","affiliation":"Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic","funding":"Supported by grant 17-20065S of the Czech Science Foundation."}],"copyrightYear":"2019","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.IPEC.2018.7","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.4230\/LIPIcs.ICALP.2017.68","http:\/\/arxiv.org\/abs\/1805.12453"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6318","volumeNumber":115,"name":"13th International Symposium on Parameterized and Exact Computation (IPEC 2018)","dateCreated":"2019-02-05","datePublished":"2019-02-05","editor":[{"@type":"Person","name":"Paul, Christophe","givenName":"Christophe","familyName":"Paul"},{"@type":"Person","name":"Pilipczuk, Michal","givenName":"Michal","familyName":"Pilipczuk"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article11473","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6318"}}}