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Min-Max Connected Multiway Cut

Hans Raj Tiwary, Petr Kolman

Abstract

We introduce a variant of the multiway cut that we call the min-max connected multiway cut. Given a graph $G=(V,E)$ and a set $Γ\subseteq V$ of $t$ terminals, partition $V$ into $t$ parts such that each part is connected and contains exactly one terminal; the objective is to minimize the maximum weight of the edges leaving any part of the partition. This problem is a natural modification of the standard multiway cut problem and it differs from it in two ways: first, the cost of a partition is defined to be the maximum size of the boundary of any part, as opposed to the sum of all boundaries, and second, the subgraph induced by each part is required to be connected. Although the modified objective function has been considered before in the literature under the name min-max multiway cut, the requirement on each component to be connected has not been studied as far as we know. We show various hardness results for this problem, including a proof of weak NP-hardness of the weighted version of the problem on graphs with tree-width two, and provide a pseudopolynomial time algorithm as well as an FPTAS for the weighted problem on trees. As a consequence of our investigation we also show that the (unconstrained) min-max multiway cut problem is NP-hard even for three terminals, strengthening the known results.

Min-Max Connected Multiway Cut

Abstract

We introduce a variant of the multiway cut that we call the min-max connected multiway cut. Given a graph and a set of terminals, partition into parts such that each part is connected and contains exactly one terminal; the objective is to minimize the maximum weight of the edges leaving any part of the partition. This problem is a natural modification of the standard multiway cut problem and it differs from it in two ways: first, the cost of a partition is defined to be the maximum size of the boundary of any part, as opposed to the sum of all boundaries, and second, the subgraph induced by each part is required to be connected. Although the modified objective function has been considered before in the literature under the name min-max multiway cut, the requirement on each component to be connected has not been studied as far as we know. We show various hardness results for this problem, including a proof of weak NP-hardness of the weighted version of the problem on graphs with tree-width two, and provide a pseudopolynomial time algorithm as well as an FPTAS for the weighted problem on trees. As a consequence of our investigation we also show that the (unconstrained) min-max multiway cut problem is NP-hard even for three terminals, strengthening the known results.
Paper Structure (15 sections, 13 theorems, 3 equations, 1 figure)

This paper contains 15 sections, 13 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Terminology
  3. Our Results
  4. Related Results
  5. Spanning Tree Congestion Problem
  6. General Graphs
  7. Min-Max Cut vs. Min-Max Connected Cut
  8. NP-hardness
  9. Trees
  10. Pseudopolynomial Algorithm
  11. FPTAS
  12. Parametrized Complexity
  13. Extension Complexity
  14. NP-hardness: Weighted Exact Min-Max Cut
  15. Concluding Remarks

Key Result

Lemma 1

Let $G=(V,E)$ be a connected graph, $\tau$ a spanning tree minimizing the spanning tree congestion, $r\in V$ any non-leaf vertex of $\tau$, and $\Gamma=\{t_1,\ldots,t_k\}\subset V$ the set of neighbors of $r$ in $\tau$. Then

Figures (1)

  • Figure 1: Min-max multiway cut vs. Min-max connected multiway cut. The fat edges are of weight two, the regular edges are of weight one. The optimal min-max cut is of cost six (the blue ovals) while the optimal min-max connected cut is of size seven (the red ovals).

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • Corollary 9
  • Theorem 10
  • Theorem 11
  • ...and 3 more