Approximating Asymmetric TSP in exponential Time

Let G be a complete directed graph with n vertices and integer edge weights in range [0,M ]. It is well known that an optimal Traveling Salesman Problem (TSP) in G can be solved in 2 time and space (all bounds are given within a polynomial factor of the input length, i.e., poly(n, logM)) and this is still the fastest known algorithm. If we allow a polynomial space only, then the best known algorithm has running time 4n. For TSP with bounded weights there is an algorithm with 1.657 ·M running time. It is a big challenge to develop an algorithm with 2 time and polynomial space. Also, it is well-known that TSP cannot be approximated within any polynomial time computable function unless P=NP. In this short note we propose a very simple algorithm that, for any 0 < ε < 1, finds (1+ε)-approximation to asymmetric TSP in 2nε−1 time and ε−1 · poly(n, logM) space. Thereby, for any fixed ε, the algorithm needs 2 steps and polynomial space to compute (1 + ε)-approximation.