Min-Max Connected Multiway Cut

2026-02-14

AI Summary

This study addresses a novel variant of the classical multiway cut problem, termed the min-max multiway cut with connectivity constraints: given a graph and a set of designated terminals, the goal is to partition the vertex set into connected components such that each component contains exactly one terminal and the maximum weight of edges cut across all components is minimized. The authors establish that this problem is weakly NP-hard even on graphs of treewidth two and strengthen the known NP-hardness result for the three-terminal case without connectivity constraints. For the weighted version on trees, they devise a pseudo-polynomial time algorithm and construct a fully polynomial-time approximation scheme (FPTAS), while also establishing several computational lower bounds for various special cases.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $\Gamma\subseteq V$ 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.

Problem

• min-max
• connected
• multiway cut
• graph partitioning
• NP-hardness

Innovation

• min-max connected multiway cut
• connected partition
• NP-hardness
• FPTAS
• tree-width