# The Complexity of Max-Min $k$-Partitioning

## TL;DR

This paper establishes the computational complexity of the max-min k-partition problem on weighted graphs, showing it is complete for the class ^P, even in restricted cases, thus highlighting its inherent computational difficulty.

## Contribution

It proves the max-min k-partition problem is ^P-complete for various cases, extending known hardness results for related graph partition problems.

## Key findings

- The problem is ^P-complete for k=2 with arbitrary weights.
- The problem remains ^P-complete for k=3 with non-negative weights.
- This complexity matches known results for MaxCut and Min-3-Cut problems.

## Abstract

In this paper we study a max-min $k$-partition problem on a weighted graph, that could model a robust $k$-coalition formation. We settle the computational complexity of this problem as complete for class $\Sigma_2^P$. This hardness holds even for $k=2$ and arbitrary weights, or $k=3$ and non-negative weights, which matches what was known on \textsc{MaxCut} and \textsc{Min-3-Cut} one level higher in the polynomial hierarchy.