The train marshalling problem Original Research Article

The problem considered in this paper arose in connection with the rearrangement of railroad cars in China. In terms of sequences the problem reads as follows: Train Marshalling Problem: Given a partition S of {1,…,n} into disjoint sets S1,…,St, find the smallest number k=K(S) so that there exists a permutation p(1),…,p(t) of {1,…,t} with the property: The sequence of numbers 1,2,…,n,1,2,…,n,…,1,2,…,n where the interval 1,2,…,n is repeated k times contains all the elements of Sp(1), then all elements of Sp(2),… , etc., and finally all elements of Sp(t). The aim of this paper is to show that the decision problem: “Given numbers n,k and a partition S of {1,2,…,n}, is K(S)⩽k?” is NP-complete. In light of this, we give a general upper bound for K(S) in terms of n.