Efficient local search algorithms for the linear ordering problem



Given a directed graph with n vertices, m edges and costs on the edges, the linear ordering problem consists of finding a permutation π of the vertices so that the total cost of the reverse edges is minimized. We present two local search algorithms, named LIST and TREE, for the neighborhood of the insert move, which can handle larger instances than existing methods. LIST is simpler and can search the whole neighborhood in O(m) time and TREE performs the neighborhood search in O(n+Δlog Δ) time, where Δ represents the maximum vertex degree. Computational experiments show good results for sparse instances using LIST, while TREE presents the best results independent of the density of the instance.