Proportionally dense subgraph of maximum size: complexity and approximation

We define a proportionally dense subgraph (PDS) as an induced subgraph of a graph with the property that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the graph. We prove that the problem of finding a PDS of maximum size is APX-hard on split graphs, and NP-hard on bipartite graphs. We also show that deciding if a PDS is inclusion-wise maximal is co-NP-complete on bipartite graphs. Nevertheless, we present a simple polynomial-time $(2-\frac{2}{\mathrm{\Delta }+1})$-approximation algorithm for the problem, where $\mathrm{\Delta }$ is the maximum degree of the graph. Finally, we show that all Hamiltonian cubic graphs with $n$ vertices (except two) have a PDS of size $\lfloor \frac{2n+1}{3}\rfloor$, which we prove to be an upper bound on the size of a PDS in cubic graphs.