A Parameterized Complexity View on Collapsing k-Cores
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.
r-Degenerate Vertex Deletion
Feedback Vertex Set
Fixed-Parameter Tractability
Kernelization Lower Bounds
Graph Algorithms
Social Network Analysis
Theory of computation~Graph algorithms analysis
Theory of computation~Parameterized complexity and exact algorithms
7:1-7:14
Regular Paper
A full version is available at [Junjie Luo et al., 2018], https://arxiv.org/abs/1805.12453.
Junjie
Luo
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
Supported by CAS-DAAD Joint Fellowship Program for Doctoral Students of UCAS.
Hendrik
Molter
Hendrik Molter
Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Berlin, Germany
Supported by the DFG, project MATE (NI 369/17).
Ondrej
Suchý
Ondrej Suchý
Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic
Supported by grant 17-20065S of the Czech Science Foundation.
10.4230/LIPIcs.IPEC.2018.7
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Junjie Luo, Hendrik Molter, and Ondřej Suchý
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