Mixed Integer Linear Programming Based Large Neighborhood Search Approaches for the Directed Feedback Vertex Set Problem

A directed feedback vertex set (DFVS) of a directed graph is a subset of vertices whose removal makes the graph acyclic. Finding a DFVS of minimum cardinality is the goal of the directed feedback vertex set problem, an NP-hard combinatorial optimization problem. We first consider two mixed integer linear programming (MILP) models for this problem, which, when solved with Gurobi, are effective on graphs of small to medium complexity but do not scale well to large instances. Aiming at better scalability and higher robustness over a large variety of graphs, we investigate a large neighborhood search (LNS) in which a destroy operator removes randomly chosen nodes from an incumbent DFVS and one of the MILP models is used for repair. Regarding the destroy operator, finding a best degree of destruction is challenging. A main contribution lies in proposing several selection strategies for this parameter as well as a strategy for choosing the more promising MILP model for repair. We evaluate the performance of the MILP models and different LNS variants on benchmark instances and compare the approaches to each other as well as to state-of-the-art procedures. Results show that our LNS variants yield clearly better solutions on average than standalone MILP solving. Even though our approaches cannot outperform the state-of-the-art, we gain valuable insights on beneficially configuring such a MILP-based LNS.