Fixed-Parameter Tractability of Hedge Cut

In the Hedge Cut problem, the edges of a graph are partitioned into groups called hedges, and the question is what is the minimum number of hedges to delete to disconnect the graph. Ghaffari, Karger, and Panigrahi [SODA 2017] showed that Hedge Cut can be solved in quasipolynomial-time, raising the hope for a polynomial time algorithm. Jaffke, Lima, Masar\'ik, Pilipczuk, and Souza [SODA 2023] complemented this result by showing that assuming the Exponential Time Hypothesis (ETH), no polynomial-time algorithm exists. In this paper, we show that Hedge Cut is fixed-parameter tractable parameterized by the solution size $\ell$ by providing an algorithm with running time $\binom{O(\log n) + \ell}{\ell} \cdot m^{O(1)}$, which can be upper bounded by $c^{\ell} \cdot (n+m)^{O(1)}$ for any constant $c>1$. This running time captures at the same time the fact that the problem is quasipolynomial-time solvable, and that it is fixed-parameter tractable parameterized by $\ell$. We further generalize this algorithm to an algorithm with running time $\binom{O(k \log n) + \ell}{\ell} \cdot n^{O(k)} \cdot m^{O(1)}$ for Hedge $k$-Cut.