# Decomposition of Trees and Paths via Correlation

## TL;DR

This paper investigates the complexity of clustering trees based on pairwise costs, revealing NP-hardness in general and star trees, while providing a complete polyhedral characterization for paths, linking to combinatorial optimization.

## Contribution

It characterizes the polyhedral structure of the clustering problem on trees and paths, including a complete TDI description for paths, advancing understanding of related combinatorial formulations.

## Key findings

- NP-hardness for general and star trees
- Polynomial solvability for paths
- Complete TDI description of the lifted multicut polytope for paths

## Abstract

We study the problem of decomposing (clustering) a tree with respect to costs attributed to pairs of nodes, so as to minimize the sum of costs for those pairs of nodes that are in the same component (cluster). For the general case and for the special case of the tree being a star, we show that the problem is NP-hard. For the special case of the tree being a path, this problem is known to be polynomial time solvable. We characterize several classes of facets of the combinatorial polytope associated with a formulation of this clustering problem in terms of lifted multicuts. In particular, our results yield a complete totally dual integral (TDI) description of the lifted multicut polytope for paths, which establishes a connection to the combinatorial properties of alternative formulations such as set partitioning.