On the Metric $s$--$t$ Path Traveling Salesman Problem

We study the metric $s$--$t$ path traveling salesman problem (TSP). An, Kleinberg, and Shmoys [Proceedings of the 44th ACM Symposium on Theory of Computing, 2012, pp. 875--886] improved on the long-standing $\frac{5}{3}$-approximation factor and presented an algorithm that achieves an approximation factor of $\frac{1+\sqrt{5}}{2}\approx1.61803$. Later, Sebö [Proceedings of the 16th Conference on Integer Programming and Combinatorial Optimization, 2013, pp. 362--374] further improved the approximation factor to $\frac{8}{5}$. We present a simple, self-contained analysis that unifies both results; our main contribution is a unified correction vector. Additionally, we compare two different linear programming (LP) relaxations of the $s$--$t$ path TSP, namely, the path version of the Held--Karp LP relaxation for the TSP and a weaker LP relaxation, and we show that both LPs have the same (fractional) optimal value. Also, we show that the minimum cost of integral solutions of the two LPs are within a factor of $\frac{3}{2}$ of each other. Furthermore, we prove that a half-integral solution of the stronger LP relaxation of cost $c$ can be rounded to an integral solution of cost at most $\frac{3}{2}c$. Finally, we give an example that presents obstructions to two natural methods that aim for an approximation factor of $\frac{3}{2}$.