Faster Approximation Schemes for k-TSP and k-MST in the Euclidean Space
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In the Euclidean k-TSP (resp. Euclidean k-MST), we are given n points in the d-dimensional Euclidean space (for any fixed constant d≥ 2) and a positive integer k, the goal is to find a shortest tour visiting at least k points (resp. a minimum tree spanning at least k points). We give approximation schemes for both Euclidean k-TSP and Euclidean k-MST in time 2^O(1/ε^d-1)· n ·(log n)^d· 4^d. This improves the running time of the previous approximation schemes due to Arora [J. ACM 1998] and Mitchell [SICOMP 1999]. Our algorithms can be derandomized by increasing the running time by a factor O(n^d). In addition, our algorithm for Euclidean k-TSP is Gap-ETH tight, given the matching Gap-ETH lower bound due to Kisfaludi-Bak, Nederlof, and Węgrzycki [FOCS 2021].
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