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The multiple traveling salesman problem (mTSP) is a well-known NP-hard problem with numerous real-world applications. In particular, this work addresses MinMax mTSP, where the objective is to minimize the max tour length (sum of Euclidean distances) among all agents. The mTSP is normally considered as a combinatorial optimization problem, but due to its computational complexity, search-based exact and heuristic algorithms become inefficient as the number of cities increases. Encouraged by the recent developments in deep reinforcement learning (dRL), this work considers the mTSP as a cooperative task and introduces a decentralized attention-based neural network method to solve the MinMax mTSP, named DAN. In DAN, agents learn fully decentralized policies to collaboratively construct a tour, by predicting the future decisions of other agents. Our model relies on the Transformer architecture, and is trained using multi-agent RL with parameter sharing, which provides natural scalability to the numbers of agents and cities.We experimentally demonstrate our model on small-to largescale mTSP instances, which involve 50 to 1000 cities and 5 to 20 agents, and compare against state-of-the-art baselines. For small-scale problems (fewer than 100 cities), DAN is able to closely match the performance of the best solver available (OR Tools, a meta-heuristic solver) given the same computation time budget. In larger-scale instances, DAN outperforms both conventional and dRL-based solvers, while keeping computation times low, and exhibits enhanced collaboration among agents.

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TiO2 nanocrystals were prepared by a controlled hydrolysis procedure at room temperature. The effect of V-doping, N-doping and V/N codoping on the lattice parameters and magnetic properties of TiO2 nanocrystals was investigated by means of X-ray diffraction, high-resolution transmission electron microscopy, X-ray photoelectron spectroscopy and vibration sample magnetometry. Doping performed at room temperature causes the expansion of lattice parameters. Undoped and doped TiO2 nanocrystals show room-temperature ferromagnetism. A monotonic correlation between saturation magnetization and the ratio of the lattice parameter c to a (c/a) was observed. Saturation magnetization of TiO2 nanocrystals increases with the value of c/a.