  We study the problem of finding and characterizing subgraphs with small
\textit{bipartiteness ratio}. We give a bicriteria approximation algorithm
\verb|SwpDB| such that if there exists a subset $S$ of volume at most $k$ and
bipartiteness ratio $\theta$, then for any $0<\epsilon<1/2$, it finds a set
$S'$ of volume at most $2k^{1+\epsilon}$ and bipartiteness ratio at most
$4\sqrt{\theta/\epsilon}$. By combining a truncation operation, we give a local
algorithm \verb|LocDB|, which has asymptotically the same approximation
guarantee as the algorithm \verb|SwpDB| on both the volume and bipartiteness
ratio of the output set, and runs in time
$O(\epsilon^2\theta^{-2}k^{1+\epsilon}\ln^3k)$, independent of the size of the
graph. Finally, we give a spectral characterization of the small dense
bipartite-like subgraphs by using the $k$th \textit{largest} eigenvalue of the
Laplacian of the graph.
