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A Parameterized Complexity View on Collapsing k-Cores

Authors Junjie Luo, Hendrik Molter, Ondrej Suchý

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Author Details

Junjie Luo
  • 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
Hendrik Molter
  • Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Berlin, Germany
Ondrej Suchý
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic

Funding

  • Luo, Junjie: Supported by CAS-DAAD Joint Fellowship Program for Doctoral Students of UCAS.
  • Molter, Hendrik: Supported by the DFG, project MATE (NI 369/17).
  • Suchý, Ondrej: Supported by grant 17-20065S of the Czech Science Foundation.

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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) https://doi.org/10.4230/LIPIcs.IPEC.2018.7

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. 
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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • r-Degenerate Vertex Deletion
  • Feedback Vertex Set
  • Fixed-Parameter Tractability
  • Kernelization Lower Bounds
  • Graph Algorithms
  • Social Network Analysis

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