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Multiway Cuts with a Choice of Representatives

Kristóf Bérczi, Tamás Király, Daniel P. Szabo

TL;DR

The paper studies generalized Multiway Cut variants where terminals are chosen as representatives from candidate sets $T_1,\dots,T_q$ and analyzes several separation objectives. It introduces Lifted Cut (LIFT-LP), extending the CKR relaxation to preserve approximation guarantees and enabling $\alpha$-approximation results for fixed $q$ in several problems, along with a $2$-approximation for the Single-to-All/Single-to-Single case. For general $q$, the work proves $o(\log q)$-inapproximability in representative-dependent cases and provides structural results via gammoids on trees and Gomory–Hu trees for general graphs. The results connect representative-based variants to Steiner Multicut and metric-labeling relaxations, offering both hardness and algorithmic bridges and expanding the applicability of CKR-based techniques to a broader class of labeling and cut problems. Overall, the framework generalizes classical multiway cut methods to a richer family of problems with preserved approximation guarantees and clear algorithmic strategies.

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.

Multiway Cuts with a Choice of Representatives

TL;DR

The paper studies generalized Multiway Cut variants where terminals are chosen as representatives from candidate sets and analyzes several separation objectives. It introduces Lifted Cut (LIFT-LP), extending the CKR relaxation to preserve approximation guarantees and enabling -approximation results for fixed in several problems, along with a -approximation for the Single-to-All/Single-to-Single case. For general , the work proves -inapproximability in representative-dependent cases and provides structural results via gammoids on trees and Gomory–Hu trees for general graphs. The results connect representative-based variants to Steiner Multicut and metric-labeling relaxations, offering both hardness and algorithmic bridges and expanding the applicability of CKR-based techniques to a broader class of labeling and cut problems. Overall, the framework generalizes classical multiway cut methods to a richer family of problems with preserved approximation guarantees and clear algorithmic strategies.

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} . 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 , 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 , we show -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 -approximation algorithm for the case where we need to choose a single representative from each candidate set. This is a generalization of the -approximation for k-cut, and we can solve it by relating the tree case to optimization over a gammoid.
Paper Structure (16 sections, 19 theorems, 3 equations, 3 figures, 1 table, 7 algorithms)

This paper contains 16 sections, 19 theorems, 3 equations, 3 figures, 1 table, 7 algorithms.

Table of Contents

  1. Introduction
  2. Previous work
  3. Our results
  4. Techniques
  5. Structure of the paper
  6. Background
  7. The CKR Relaxation and Rounding Methods
  8. Other Relevant Tools
  9. Lifted Cuts
  10. Single-to-All Problem
  11. Single-to-Single Problem
  12. Fixed-to-Single, Some-to-Single, Some-to-Some, and Some-to-All Problems
  13. Hardness of approximation
  14. Fixed $q$
  15. Hardness Proofs for Fixed-to-Single, Some-to-Single, and Some-to-All
  16. ...and 1 more sections

Key Result

Theorem 1

For any fixed $0<\alpha<1$, Hitting Set cannot be approximated in polynomial time within a factor of $(1-\alpha)\ln N$ on inputs of size $N$, unless $\mathrm{P}=\mathrm{NP}$.

Figures (3)

  • Figure 1: An example of the original CKR relaxation in relation to our extended LIFT-LP on the case for $t_i-T_j$. The point colors represent the different candidate sets.
  • Figure 2: A picture of the reduction for Fixed-to-Single from Hitting Set.
  • Figure 3: A picture of the reduction from Fixed-to-Single to Some-to-Single.

Theorems & Definitions (19)

  • Theorem 1: Dinurmoshkovitz
  • Theorem 2: kcut
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 9
  • Theorem 10
  • Theorem 11
  • ...and 9 more