Parameterized Complexity of the Anchored k-Core Problem for Directed Graphs

TL;DR

This paper extends the anchored k-core problem to directed graphs, analyzing its computational complexity and fixed parameter tractability under various constraints and parameters.

Contribution

It introduces the directed version of the problem and provides new complexity and algorithmic results, including NP-completeness and fixed parameter tractability.

Findings

NP-complete for planar directed acyclic graphs with max degree at most k+2

FPT when parameterized by core size p for k=1

W[1]-hard for k>=2

Abstract

Bhawalkar, Kleinberg, Lewi, Roughgarden, and Sharma [ICALP 2012] introduced the Anchored k-Core problem, where the task is for a given graph G and integers b, k, and p to find an induced subgraph H with at least p vertices (the core) such that all but at most b vertices (called anchors) of H are of degree at least k. In this paper, we extend the notion of k-core to directed graphs and provide a number of new algorithmic and complexity results for the directed version of the problem. We show that - The decision version of the problem is NP-complete for every k>=1 even if the input graph is restricted to be a planar directed acyclic graph of maximum degree at most k+2. - The problem is fixed parameter tractable (FPT) parameterized by the size of the core p for k=1, and W[1]-hard for k>=2. - When the maximum degree of the graph is at most \Delta, the problem is FPT parameterized by p+\Delta if k>= \Delta/2.