On the Stability of Approximation for Hamiltonian Path Problems

We consider the problem of finding a cheapest Hamiltonian path of a complete graph satisfying a relaxed triangle inequality, i.e., such that for some parameter $\beta > 1$, the edge costs satisfy the inequality $c(\{x,y\})\le\beta(c(\{x,z\})+c(\{z,y\}))$ for every triple of vertices $x$, $y$, $z$. There are three variants of this problem, depending on the number of prespecified endpoints: zero, one, or two. For metric graphs, there exist approximation algorithms, with approximation ratio $\frac{3}{2}$ for the first two variants and $\frac{5}{3}$ for the latter one, respectively. Using results on the approximability of the Travelling Salesman Problem with input graphs satisfying the relaxed triangle inequality, we obtain for our problem approximation algorithms with ratio $\min(\beta^2+\beta,\frac{3}{2}\beta^2)$ for zero or one prespecified endpoints, and $\frac{5}{3}\beta^2$ for two endpoints.