Group for Research in Decision Analysis

# Computational study of valid inequalities for the maximum $$k$$-cut problem

## Vilmar Jefté Rodrigues De Sousa, Miguel F. Anjos, and Sébastien Le Digabel

We consider the maximum $$k$$-cut problem that consists in partitioning the vertex set of a graph into $$k$$ subsets such that the sum of the weights of edges joining vertices in different subsets is maximized. We focus on identifying effective classes of inequalities to tighten the semidefinite programming relaxation. We carry out an experimental study of four classes of inequalities from the literature: clique, genclique, wheel and biwheel. We considered 10 combinations of these classes and tested them on both dense and sparse instances for $$k \in \{3,4,5,7\}$$. Our computational results suggest that the biwheel and wheel are the strongest inequalities for $$k=3$$, and that for $$k \in \{4,5,7\}$$ the wheel inequalities are the strongest by far. Furthermore, we observe an improvement in the performance for all choices of $$k$$ when both biwheel and wheel are used, at the cost of 72% more CPU time on average when compared with using only one of them.

, 24 pages