Approximation algorithms for the k-depots Hamiltonian path problem

We consider a multiple-depots extension of the classic Hamiltonian path problem where k salesmen are initially located at different depots. To the best of our knowledge, no algorithm for this problem with a constant approximation ratio has been previously proposed, except for some special cases. We present a polynomial algorithm with a tight approximation ratio of max {3/2, 2 - 1/k} for arbitrary k >= 1, and an algorithm with approximation ratio 5/3 that runs in polynomial time for fixed k. Moreover, we develop a recursive framework to improve the approximation ratio to 3/2 + epsilon. This framework is polynomial for fixed k and epsilon, and may be useful in improving the Christofides-like heuristics for other related multiple salesmen routing problems.