  We introduce and study $\ell_p$-norm-multiway-cut: the input here is an
undirected graph with non-negative edge weights along with $k$ terminals and
the goal is to find a partition of the vertex set into $k$ parts each
containing exactly one terminal so as to minimize the $\ell_p$-norm of the cut
values of the parts. This is a unified generalization of min-sum multiway cut
(when $p=1$) and min-max multiway cut (when $p=\infty$), both of which are
well-studied classic problems in the graph partitioning literature. We show
that $\ell_p$-norm-multiway-cut is NP-hard for constant number of terminals and
is NP-hard in planar graphs. On the algorithmic side, we design an $O(\log^2
n)$-approximation for all $p\ge 1$. We also show an integrality gap of
$\Omega(k^{1-1/p})$ for a natural convex program and an
$O(k^{1-1/p-\epsilon})$-inapproximability for any constant $\epsilon>0$
assuming the small set expansion hypothesis.
