Analysis of Christofides' heuristic : some paths are more difficult than cycles

For the traveling salesman problem in which the distances satisfy the triangle inequality, Christofides' heuristic produces a tour whose length is guaranteed to be less than 3/2 times the optimum tour length. We investigate the performance of appropriate modifications of this heuristic for the problem of finding a shortest Hamiltonian path. There are three variants of this problem, depending on the number of prespecified endpoints: zero, one, or two. It is not hard to see that, for the first two problems, the worst-case performance ratio of a Christofides-like heuristic is still 3/2. For the third case, we show that the ratio is 5/3 and that this bound is tight.