  In this paper, we address the challenge of differential privacy in the
context of graph cuts, specifically focusing on the minimum $k$-cut and
multiway cut problems. We introduce edge-differentially private algorithms that
achieve nearly optimal performance for these problems.
  For the multiway cut problem, we first provide a private algorithm with a
multiplicative approximation ratio that matches the state-of-the-art
non-private algorithm. We then present a tight information-theoretic lower
bound on the additive error, demonstrating that our algorithm on weighted
graphs is near-optimal for constant $k$. For the minimum $k$-cut problem, our
algorithms leverage a known bound on the number of approximate $k$-cuts,
resulting in a private algorithm with optimal additive error $O(k\log n)$ for
fixed privacy parameter. We also establish a information-theoretic lower bound
that matches this additive error. Additionally, we give an efficient private
algorithm for $k$-cut even for non-constant $k$, including a polynomial-time
2-approximation with an additive error of $\widetilde{O}(k^{1.5})$.
