We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an n-vertex,
k-regular graph, the algorithm computes a tour of length at most
(1+lnk−O(1)7)n, with high probability, in O(nklogk) time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with O(n/logk) cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
23+O(k1) factor approximation algorithm
running in time O(nk).Comment: 12 page