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Multiway cuts

In the paper Multiway Cuts with a Choice of Representatives, appearing at MFCS 2024, we study several generalizations of multiway cut where the terminals can be chosen as representatives from sets of candidates $T_1,\dots,T_q$. In this setting, one is allowed to choose these representatives so that the minimum-weight cut separating these sets via their representatives is as small as possible. We distinguish different cases depending on (A) whether the representative of a candidate set has to be separated from the other candidate sets completely or only from the representatives, and (B) whether there is a single representative for each candidate set or the choice of representative is independent for each pair of candidate sets. For fixed $q$, we give approximation algorithms for each of these problems that match the best known approximation guarantee for multiway cut. Our technical contribution is a new extension of the CKR relaxation that preserves approximation guarantees. For general $q$, we show $o(\log q)$-inapproximability for all cases where the choice of representatives may depend on the pair of candidate sets, as well as for the case where the goal is to separate a fixed node from a single representative from each candidate set. As a positive result, we give a 2-approximation algorithm for the case where we need to choose a single representative from each candidate set. This is a generalization of the $(2−2/k)$-approximation for $k$-cut, and we can solve it by relating the tree case to optimization over a gammoid.