We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a
node-disjoint cycle cover in a directed graph with minimum interaction cost
between successive arcs. We derive several semidefinite programming (SDP)
relaxations and use facial reduction to make these strictly feasible. We
investigate a nontrivial relationship between the transformation matrix used in
the reduction and the structure of the graph, which is exploited in an
efficient algorithm that constructs this matrix for any instance of the
problem. To solve our relaxations, we propose an algorithm that incorporates an
augmented Lagrangian method into a cutting plane framework by utilizing
Dykstra's projection algorithm. Our algorithm is suitable for solving SDP
relaxations with a large number of cutting planes. Computational results show
that our SDP bounds and our efficient cutting plane algorithm outperform other
QCCP bounding approaches from the literature. Finally, we provide several
SDP-based upper bounding techniques, among which a sequential Q-learning method
that exploits a solution of our SDP relaxation within a reinforcement learning
environment