Computing Densest $k$-Subgraph with Structural Parameters

TL;DR

This paper investigates the computational complexity of the Densest k-Subgraph problem, demonstrating fixed parameter tractability with respect to certain structural graph parameters and providing a 2-approximation algorithm based on twin cover number.

Contribution

It establishes fixed parameter tractability of Densest k-Subgraph for various structural parameters and introduces a 2-approximation algorithm leveraging twin cover number.

Findings

FPT results for neighborhood diversity, block deletion, distance-hereditary, and cograph deletion numbers.

A 2-approximation algorithm with runtime depending on twin cover number.

Enhanced understanding of the problem's complexity relative to graph structural parameters.

Abstract

\textsc{Densest $k$-Subgraph} is the problem to find a vertex subset $S$ of size $k$ such that the number of edges in the subgraph induced by $S$ is maximized. In this paper, we show that \textsc{Densest $k$-Subgraph} is fixed parameter tractable when parameterized by neighborhood diversity, block deletion number, distance-hereditary deletion number, and cograph deletion number, respectively. Furthermore, we give a $2$-approximation $2^{\tc(G)/2}n^{O(1)}$-time algorithm where $\tc(G)$ is the twin cover number of an input graph $G$.