The paper presents a polynomial-time approximation scheme (PTAS) for the Euclidean Traveling Salesman Problem (TSP) in fixed dimensions. For any fixed \( c > 1 \) and given \( n \) nodes in \( \mathbb{R}^2 \), the randomized algorithm finds a \((1 + 1/c)\)-approximation to the optimal tour in \( O(n (\log n)^{O(c)}) \) time. For nodes in \( \mathbb{R}^d \), the running time increases to \( O(n (\log n)^{O(c/d)}) \). The algorithm can be derandomized but at a cost of \( O(n^d) \). The paper also provides similar PTASs for other NP-hard Euclidean problems, such as Minimum Steiner Tree, \( k \)-TSP, and \( k \)-MST, and an efficient PTAS for Euclidean Min-Cost Matching. The techniques used are applicable to geometric problems measured by any Minkowski norm and have simple parallel implementations. The main idea is to recursively partition the plane using a randomized variant of the quadtree, ensuring that a \((1 + 1/c)\)-approximate tour crosses each partition line at most \( r = O(c) \) times. This allows for dynamic programming to find the optimal tour in \( n \cdot (\log n)^{\tilde{O}(c)} \) time. The paper discusses the practicality of the algorithm and its potential contributions to understanding local-exchange heuristics for Euclidean instances.The paper presents a polynomial-time approximation scheme (PTAS) for the Euclidean Traveling Salesman Problem (TSP) in fixed dimensions. For any fixed \( c > 1 \) and given \( n \) nodes in \( \mathbb{R}^2 \), the randomized algorithm finds a \((1 + 1/c)\)-approximation to the optimal tour in \( O(n (\log n)^{O(c)}) \) time. For nodes in \( \mathbb{R}^d \), the running time increases to \( O(n (\log n)^{O(c/d)}) \). The algorithm can be derandomized but at a cost of \( O(n^d) \). The paper also provides similar PTASs for other NP-hard Euclidean problems, such as Minimum Steiner Tree, \( k \)-TSP, and \( k \)-MST, and an efficient PTAS for Euclidean Min-Cost Matching. The techniques used are applicable to geometric problems measured by any Minkowski norm and have simple parallel implementations. The main idea is to recursively partition the plane using a randomized variant of the quadtree, ensuring that a \((1 + 1/c)\)-approximate tour crosses each partition line at most \( r = O(c) \) times. This allows for dynamic programming to find the optimal tour in \( n \cdot (\log n)^{\tilde{O}(c)} \) time. The paper discusses the practicality of the algorithm and its potential contributions to understanding local-exchange heuristics for Euclidean instances.