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

Abstract

We dedicate this paper to Tamás Terlaky on the occasion of his 60 th birthday.

Improving the linear relaxation of maximum k-cut with semidefinite-based constraints

“…The matrix constraint (9) has an infinite number of rows. By replacing (4) by (9) in MkC-SDP we obtain the LSIP formulation of SDP.…”

“…Separation routines are algorithms that search for violations of a given family of valid inequalities in a relaxed solution. In this section we present separation routines for some inequalities studied in [9], for Constraint (3) in the SDP formulation, and for Constraint (10) proposed in this work.…”

“…Five of these families of constraints are explored computationally in [9], where heuristic and exact methods are proposed. In this work, we replicate the best separation routines of [9] for the following families of inequalities:…”

“…• Bicycle wheel: genetic algorithm. In [9] the authors concluded that in practice, wheel and triangle are the best inequalities. Hence, we prioritize these two families of inequalities in our ranking algorithm (see Section 3.1.4).…”

“…As shown in [9], the inclusion of all the violated inequalities in a CPA iteration can be computationally impractical. It is better to rank the violated inequalities and append only those that are most violated.…”