  We prove an approximate max-multiflow min-multicut theorem for bounded
treewidth graphs. In particular, we show the following: Given a treewidth-$r$
graph, there exists a (fractional) multicommodity flow of value $f$, and a
multicut of capacity $c$ such that $ f \leq c \leq \mathcal{O}(\ln (r+1)) \cdot
f$. It is well known that the multiflow-multicut gap on an $r$-vertex (constant
degree) expander graph can be $\Omega(\ln r)$, and hence our result is tight up
to constant factors. Our proof is constructive, and we also obtain a polynomial
time $\mathcal{O}(\ln (r+1))$-approximation algorithm for the minimum multicut
problem on treewidth-$r$ graphs. Our algorithm proceeds by rounding the optimal
fractional solution to the natural linear programming relaxation of the
multicut problem. We introduce novel modifications to the well-known region
growing algorithm to facilitate the rounding while guaranteeing at most a
logarithmic factor loss in the treewidth.
