Large Isolating Cuts Shrink the Multiway Cut

We propose a preprocessing algorithm for the multiway cut problem that establishes its polynomial kernelizability when the difference between the parameter $k$ and the size of the smallest isolating cut is at most $log(k)$. To the best of our knowledge, this is the first progress towards kernelization of the multiway cut problem. We pose two open questions that, if answered affirmatively, would imply, combined with the proposed result, unconditional polynomial kernelizability of the multiway cut problem.