# Polyhedral Properties of the Induced Cluster Subgraphs

## TL;DR

This paper studies the polyhedral structure of the maximum independent union of cliques problem, deriving new inequalities, describing the polytope for special graphs, and testing cutting-plane methods computationally.

## Contribution

It introduces new facet-defining inequalities for the IUC polytope and provides a complete description for certain graph classes, advancing optimization techniques for the problem.

## Key findings

- Derived several families of facet-defining inequalities.
- Provided a complete polytope description for specific graph classes.
- Demonstrated the effectiveness of cutting-plane algorithms in computational experiments.

## Abstract

A cluster graph is a graph whose every connected component is a complete graph. Given a simple undirected graph $G$, a subset of vertices inducing a cluster graph is called an independent union of cliques (IUC), and the IUC polytope associated with $G$ is defined as the convex hull of the incidence vectors of all IUCs in the graph. The {\sc Maximum IUC} problem, which is to find a maximum-cardinality IUC in a graph, finds applications in network-based data analysis. In this paper, we derive several families of facet-defining valid inequalities for the IUC polytope. We also give a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problem for most of the considered families of valid inequalities and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the {\sc Maximum IUC} problem through computational experiments.