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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 designcombinatorial 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 $k$-subgraphin $G$ (which is not necessarily connected). These particularly provide thefirst non-trivial approximations for the densest connected $k$-subgraph problemon general graphs.

Finding Connected Dense $k$-Subgraphs