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The MAX-MIN Ant System (MMAS) is one of the best-known Ant Colony Optimization (ACO) algorithms proven to be efficient at finding satisfactory solutions to many difficult combinatorial optimization problems. The slow-down in Moore's law, and the availability of graphics processing units (GPUs) capable of conducting general-purpose computations at high speed, has sparked considerable research efforts into the development of GPU-based ACO implementations. In this paper, we discuss a range of novel ideas for improving the GPU-based parallel MMAS implementation, allowing it to better utilize the computing power offered by two subsequent Nvidia GPU architectures. Specifically, based on the weighted reservoir sampling algorithm we propose a novel parallel implementation of the node selection procedure, which is at the heart of the MMAS and other ACO algorithms. We also present a memory-efficient implementation of another key-component -- the tabu list structure -- which is used in the ACO's solution construction stage. The proposed implementations, combined with the existing approaches, lead to a total of six MMAS variants, which are evaluated on a set of Traveling Salesman Problem (TSP) instances ranging from 198 to 3,795 cities. The results show that our MMAS implementation is competitive with state-of-the-art GPU-based and multi-core CPU-based parallel ACO implementations: in fact, the times obtained for the Nvidia V100 Volta GPU were up to 7.18x and 21.79x smaller, respectively. The fastest of the proposed MMAS variants is able to generate over 1 million candidate solutions per second when solving a 1,002-city instance. Moreover, we show that, combined with the 2-opt local search heuristic, the proposed parallel MMAS finds high-quality solutions for the TSP instances with up to 18,512 nodes.