  Recent papers on approximation algorithms for the traveling salesman problem
(TSP) have given a new variant on the well-known Christofides' algorithm for
the TSP, called the Best-of-Many Christofides' algorithm. The algorithm
involves sampling a spanning tree from the solution the standard LP relaxation
of the TSP, subject to the condition that each edge is sampled with probability
at most its value in the LP relaxation. One then runs Christofides' algorithm
on the tree by computing a minimum-cost matching on the odd-degree vertices in
the tree, and shortcutting the resulting Eulerian graph to a tour. In this
paper we perform an experimental evaluation of the Best-of-Many Christofides'
algorithm to see if there are empirical reasons to believe its performance is
better than that of Christofides' algorithm. Furthermore, several different
sampling schemes have been proposed; we implement several different schemes to
determine which ones might be the most promising for obtaining improved
performance guarantees over that of Christofides' algorithm. In our
experiments, all of the implemented methods perform significantly better than
the Christofides' algorithm; an algorithm that samples from a maximum entropy
distribution over spanning trees seems to be particularly good, though there
are others that perform almost as well.
