A Parameterized Complexity View on Collapsing k-Cores

Junjie Luo,Ondřej Suchý,Hendrik Molter

doi:10.1007/s00224-021-10045-w

Junjie Luo, Ondřej Suchý + Show 1 more

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https://doi.org/10.1007/s00224-021-10045-w

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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. (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.

Highlights

In recent years, modelling user engagement in social networks has received substantial interest [44, 45]
We show that COLLAPSED k-CORE is fixed-parameter tractable when parameterized by the treewidth of the input graph and show that it presumably does not admit a polynomial kernel when parameterized by either the vertex cover number or the bandwidth of the input graph
The hardness of COLLAPSED k-CORE was first established by Mathieson [38] who showed that r-DEGENERATE VERTEX DELETION is NP-complete and W[P]-complete when parameterized by the budget b for all r ≥ 2 even if the input graph is already (r + 1)-degenerate and has maximum degree 2r + 1

Summary

Introduction

In recent years, modelling user engagement in social networks has received substantial interest [44, 45]. Given a stable social network, that is, a graph with minimum degree k, the departure of a user decreases the degree of her neighbors in the graph by one which might be smaller than k for some of them Following our assumption these users will abandon the network, too. From an adversarial perspective a natural question is how to maximally destabilize a competing social network platform by compelling b users to abandon the network. This problem was introduced as COLLAPSED k-CORE by Zhang et al [46] and the decision version is formally defined as follows

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