Fixed-Parameter Tractability of Hedge Cut

Abstract

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, Masarík, 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 $\left(\genfrac{}{}{0ex}{}{O(\log n)+\ell }{\ell }\right)\cdot m^{O(1)}$, 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 $\left(\genfrac{}{}{0ex}{}{O(k\log n)+\ell }{\ell }\right)\cdot n^{O(k)}\cdot m^{O(1)}$ for Hedge $k$-Cut.

12 pages, 1 figure, to appear in SODA 2025