Improved approximations for ordered TSP on near-metric graphs

Abstract

The traveling salesman problem is one of the most important problems in operations research, especially when additional precedence constraints are considered. Here, we consider the well-known variant where a linear order on k special vertices is given that has to be preserved in any feasible Hamiltonian cycle. This problem is called Ordered TSP and we consider it on input instances where the edge-cost function satisfies a β -relaxed triangle inequality, i.e. , where the length of a direct edge cannot exceed the cost of any detour via a third vertex by more than a factor of β > 1. We design two new polynomial-time approximation algorithms for this problem. The first algorithm essentially improves over the best previously known algorithm for almost all values of k and β n ≥ 11 k + 7 and β 3 4/3 , where n is the number of vertices in the graph.