TY - GEN

T1 - A parameterized complexity view on collapsing k-cores

AU - Luo, Junjie

AU - Molter, Hendrik

AU - Suchý, Ondrej

N1 - Publisher Copyright:
© Junjie Luo, Hendrik Molter, and Ondrej Suchý; licensed under Creative Commons License CC-BY.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Feedback Vertex Set

KW - Fixed-Parameter Tractability

KW - Graph Algorithms

KW - Kernelization Lower Bounds

KW - R-Degenerate Vertex Deletion

KW - Social Network Analysis

UR - http://www.scopus.com/inward/record.url?scp=85090506158&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.IPEC.2018.7

DO - 10.4230/LIPIcs.IPEC.2018.7

M3 - Conference contribution

AN - SCOPUS:85090506158

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 13th International Symposium on Parameterized and Exact Computation, IPEC 2018

A2 - Paul, Christophe

A2 - Pilipczuk, Michal

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 13th International Symposium on Parameterized and Exact Computation, IPEC 2018

Y2 - 22 August 2018 through 24 August 2018

ER -