Given a connected graph G on n vertices and a positive integer k≤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
Ω(max{n−2/5,k2/n2}) 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