  In the $k$-cut problem, we want to find the lowest-weight set of edges whose
deletion breaks a given (multi)graph into $k$ connected components. Algorithms
of Karger \& Stein can solve this in roughly $O(n^{2k})$ time. On the other
hand, lower bounds from conjectures about the $k$-clique problem imply that
$\Omega(n^{(1-o(1))k})$ time is likely needed. Recent results of Gupta, Lee \&
Li have given new algorithms for general $k$-cut in $n^{1.98k + O(1)}$ time, as
well as specialized algorithms with better performance for certain classes of
graphs (e.g., for small integer edge weights).
  In this work, we resolve the problem for general graphs. We show that the
Contraction Algorithm of Karger outputs any fixed $k$-cut of weight $\alpha
\lambda_k$ with probability $\Omega_k(n^{-\alpha k})$, where $\lambda_k$
denotes the minimum $k$-cut weight. This also gives an extremal bound of
$O_k(n^k)$ on the number of minimum $k$-cuts and an algorithm to compute
$\lambda_k$ with roughly $n^k \mathrm{polylog}(n)$ runtime. Both are tight up
to lower-order factors, with the algorithmic lower bound assuming hardness of
max-weight $k$-clique.
  The first main ingredient in our result is an extremal bound on the number of
cuts of weight less than $2 \lambda_k/k$, using the Sunflower lemma. The second
ingredient is a fine-grained analysis of how the graph shrinks -- and how the
average degree evolves -- in the Karger process.
