The traveling salesman problem is one of the most important problems in operationsresearch, especially when additional precedence constraints are considered. Here, weconsider the well-known variant where a linear order on k special vertices is giventhat has to be preserved in any feasible Hamiltonian cycle. This problem is called OrderedTSP and we consider it on input instances where the edge-cost function satisfies aβ-relaxedtriangle inequality, i.e., where the length of a direct edge cannotexceed the cost of any detour via a third vertex by more than a factor ofβ> 1. Wedesign two new polynomial-time approximation algorithms for this problem. The firstalgorithm essentially improves over the best previously known algorithm for almost allvalues of kand β<1.087889. The second algorithm gives a further improvement for2n ≥ 11k +7 and β< √34/3 , where n is the number of vertices in the graph.
