A Parameterized Complexity View on Collapsing k-Cores
Result description
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. (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. For k <= 2 we show that Collapsed k-Core is W[1]-hard when parameterized by b and in FPT when parameterized by (b + x). Furthermore, we outline 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 deletionFeedback vertex setFixed-parameter tractabilityKernelization lower boundsGraph algorithmsSocial network analysis
The result's identifiers
Result code in IS VaVaI
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
A Parameterized Complexity View on Collapsing k-Cores
Original language description
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. (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. For k <= 2 we show that Collapsed k-Core is W[1]-hard when parameterized by b and in FPT when parameterized by (b + x). Furthermore, we outline 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.
Czech name
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Czech description
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Classification
Type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
GA17-20065S: Tight Parameterized Results for Directed Connectivity Problems
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Theory of Computing Systems
ISSN
1432-4350
e-ISSN
1433-0490
Volume of the periodical
65
Issue of the periodical within the volume
8
Country of publishing house
US - UNITED STATES
Number of pages
40
Pages from-to
1243-1282
UT code for WoS article
000663468900002
EID of the result in the Scopus database
2-s2.0-85108313727
Basic information
Result type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
OECD FORD
Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Year of implementation
2021