A 1.5-Approximation for Path TSP
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We present a 1.5-approximation for the Metric Path Traveling Salesman Problem (path TSP). All recent improvements on the path TSP crucially exploit a structural property shown by An, Kleinberg, and Shmoys [Journal of the ACM, 2015], namely that narrow cuts with respect to a Held-Karp solution form a chain. We significantly deviate from these approaches by showing the benefit to deal with larger s-t cuts, even though they are much less structured. More precisely, we show that a variation of the dynamic programming idea recently introduced by Traub and Vygen [SODA, 2018] is versatile enough to deal with larger size cuts, by exploiting a seminal result of Karger on the number of near-minimum cuts. This avoids a recursive application of dynamic programming as used by Traub and Vygen, and leads to a considerable simpler algorithm avoiding an additional error term in the approximation guarantee. Because we match the still unbeaten 1.5-approximation guarantee of Christofides' algorithm for TSP, any further progress on the approximability of the path TSP will also lead to an improvement for TSP.
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