In this paper, we study several generalizations of multiway cut where the terminals can be chosen as \emph{representatives} from sets of \emph{candidates} $T_1,\ldots,T_q$. In this setting, one is allowed to choose these representatives so that the minimum-weight cut separating these sets \emph{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.

翻译：本文研究多路割的若干推广形式，其中终端节点可从候选集合$T_1,\ldots,T_q$中选取代表。在此设定下，允许通过选择代表的方式，使得分离这些集合（通过其代表）的最小权割尽可能小。我们根据以下两种情形进行区分：(A) 候选集合的代表是需要与其他候选集合完全分离，还是仅需与其他代表分离；(B) 每个候选集合是选择单一代表，还是每对候选集合的代表选择相互独立。对于固定的$q$，我们针对每个问题给出了与多路割最佳已知近似保证相匹配的近似算法。我们的技术贡献在于提出了一种能保持近似保证的CKR松弛新扩展形式。对于一般$q$，我们证明了当代表选择可能依赖于候选集合对时所有情形的$o(\log q)$不可近似性，以及需要从每个候选集合选取单一代表来分离固定节点的情形。作为积极结果，我们针对需要从每个候选集合选取单一代表的情形给出了2-近似算法。这是k-割问题$(2-2/k)$-近似解的推广形式，可通过将树形情形关联到拟阵优化问题进行求解。