Finding Connected Dense $k$-Subgraphs

TL;DR

This paper introduces approximation algorithms for finding connected subgraphs of size k with high density, providing the first non-trivial solutions for this problem in general graphs.

Contribution

It presents the first non-trivial approximation algorithms for the densest connected k-subgraph problem on general graphs.

Findings

Achieves approximation factor of (\maxig {n^{-2/5},k^2/n^2}ig) for density.

First non-trivial approximation algorithms for this problem.

Applicable to general graphs, not restricted to special cases.

Abstract

Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$ such that its density is at least a factor $\Omega(\max\{n^{-2/5},k^2/n^2\})$ of the density of the densest $k$-subgraph in $G$ (which is not necessarily connected). These particularly provide the first non-trivial approximations for the densest connected $k$-subgraph problem on general graphs.