Multiway Cuts with a Choice of Representatives

TL;DR

This paper explores generalized multiway cut problems with representative choices, providing approximation algorithms, extending relaxations, and establishing inapproximability bounds for various cases.

Contribution

It introduces new approximation algorithms and relaxations for multiway cut variants with representative choices, and proves inapproximability results for certain cases.

Findings

Approximation algorithms match the best known guarantees for fixed q.

A new extension of the CKR relaxation preserves approximation guarantees.

A 2-approximation algorithm is provided for selecting a single representative from each set.

Abstract

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.