A 4/3-approximation for TSP on cubic 3-edge-connected graphs

We consider the travelling salesman problem on metrics which can be viewed as the shortest path metric of an undirected graph with unit edge-lengths. Finding a TSP tour in such a metric is then equivalent to finding a connected Eulerian subgraph in the underlying graph. Since the length of the tour is the number of edges in this Eulerian subgraph our problem can equivalently be stated as follows: Given an undirected, unweighted graph G = (V,E) find a connected Eulerian subgraph, H = (V,E′) with the fewest edges. Note that H could be a multigraph. In this paper we consider the special case of the problem when G is 3-regular (also called cubic) and 3-edge-connected. Note that the smallest Eulerian subgraph contains at least n = |V | edges. In fact, in the shortest path metric arising out of such a graph the Held-Karp bound for the length of the TSP tour would also be n. This is because we can obtain a fractional solution to the sub-tour elimination LP (which is equivalent to the Held-Karp bound) of value n by assigning 2/3 to every edge in G. Improving the approximation ratio for metric-TSP beyond 3/2 is a long standing open problem. For the metric completion of cubic 3-edge connected graphs Gamarnik et.al. [1] obtained an algorithm with an approximation guarantee slightly better than 3/2. The main result of this paper is to improve this approximation guarantee to 4/3 by giving a polynomial time algorithm to find a connected Eulerian subgraph with at most 4n/3 edges. This matches the conjectured integrality gap for the sub-tour elimination LP for the special case of these metrics.