  We introduce the $st$-cut version the Sparsest-Cut problem, where the goal is
to find a cut of minimum sparsity among those separating two distinguished
vertices $s,t\in V$. Clearly, this problem is at least as hard as the usual
(non-$st$) version. Our main result is a polynomial-time algorithm for the
product-demands setting, that produces a cut of sparsity $O(\sqrt{\OPT})$,
where $\OPT$ denotes the optimum, and the total edge capacity and the total
demand are assumed (by normalization) to be $1$.
  Our result generalizes the recent work of Trevisan [arXiv, 2013] for the
non-$st$ version of the same problem (Sparsest-Cut with product demands), which
in turn generalizes the bound achieved by the discrete Cheeger inequality, a
cornerstone of Spectral Graph Theory that has numerous applications. Indeed,
Cheeger's inequality handles graph conductance, the special case of product
demands that are proportional to the vertex (capacitated) degrees. Along the
way, we obtain an $O(\log n)$-approximation, where $n=\card{V}$, for the
general-demands setting of Sparsest $st$-Cut.
