@MISC{Sebő12eight-fifthapproximation, author = {András Sebő}, title = {Eight-Fifth Approximation for TSP Paths}, year = {2012} }

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Abstract

We prove the approximation ratio 8/5 for the metric {s, t}-path-TSP problem, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple “Best of Many ” version of Christofides’ algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determin-ing the best Christofides {s, t}-tour out of those constructed from a family F>0 of trees having a convex combination dominated by an optimal solution x ∗ of the fractional relaxation. They give the approximation guarantee 5+1 2 for such an {s, t}-tour, which is the first improve-ment after the 5/3 guarantee of Hoogeveen’s Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13/8-approximation of shortest connected T-joins, for |T | ≥ 4. The ratio 8/5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x∗/2 in order to dominate the cost of “parity correction” for spanning trees. We partition the edge-set of each spanning tree in F>0 into an {s, t}-path (or more generally, into a T-join) and its complement, which induces a decomposition of x∗. This decomposition can be refined and then efficiently used to complete x∗/2 without using linear programming or particular properties of T, but by adding to each cut deficient for x∗/2 an individually tailored explicitly given vector, inherent in x∗. A simple example shows that the Best of Many Christofides algorithm may not find a shorter {s, t}-tour than 3/2 times the incidentally common optima of the problem and of its fractional relaxation.