Recently, neural heuristics based on deep reinforcement learning have exhibited promise in solving multi-objective combinatorial optimization problems (MOCOPs). However, they are still struggling to achieve high learning efficiency and solution quality. To tackle this issue, we propose an efficient meta neural heuristic (EMNH), in which a meta-model is first trained and then fine-tuned with a few steps to solve corresponding single-objective subproblems. Specifically, for the training process, a (partial) architecture-shared multi-task model is leveraged to achieve parallel learning for the meta-model, so as to speed up the training; meanwhile, a scaled symmetric sampling method with respect to the weight vectors is designed to stabilize the training. For the fine-tuning process, an efficient hierarchical method is proposed to systematically tackle all the subproblems. Experimental results on the multi-objective traveling salesman problem (MOTSP), multi-objective capacitated vehicle routing problem (MOCVRP), and multi-objective knapsack problem (MOKP) show that, EMNH is able to outperform the state-of-the-art neural heuristics in terms of solution quality and learning efficiency, and yield competitive solutions to the strong traditional heuristics while consuming much shorter time.

Most of existing neural methods for multi-objective combinatorial optimization (MOCO) problems solely rely on decomposition, which often leads to repetitive solutions for the respective subproblems, thus a limited Pareto set. Beyond decomposition, we propose a novel neural heuristic with diversity enhancement (NHDE) to produce more Pareto solutions from two perspectives. On the one hand, to hinder duplicated solutions for different subproblems, we propose an indicator-enhanced deep reinforcement learning method to guide the model, and design a heterogeneous graph attention mechanism to capture the relations between the instance graph and the Pareto front graph. On the other hand, to excavate more solutions in the neighborhood of each subproblem, we present a multiple Pareto optima strategy to sample and preserve desirable solutions. Experimental results on classic MOCO problems show that our NHDE is able to generate a Pareto front with higher diversity, thereby achieving superior overall performance. Moreover, our NHDE is generic and can be applied to different neural methods for MOCO.

The multiobjective combinatorial optimization problems (MOCOPs) have a wide range of real-world applications. Designing an effective algorithm has an important and practical significance. Due to the huge search space and limited time, it is generally difficult to obtain the optimal solution of this kind of problem by traditional exact and heuristic algorithms. Recently, learning-based algorithms have achieved good results in solving MOCOPs, but the quality and diversity of found solutions can be further improved. In this paper, we propose a Weight-Specific-Decoder Attention Model (WSDAM) to better approximate the whole Pareto set. It embeds a weight-adaptive layer into the decoder to concentrate on the information of different weight vectors. During the model training, the weight vector is sampled from the Dirichlet distribution, which can further strengthen the learning of boundary solutions. We evaluate our method on two classic MOCOPs, i.e., the multiobjective traveling salesman problem (MOTSP) and multiobjective capacitated vehicle routing problem (MOCVRP). The experimental results show that our proposed method outperforms current state-of-the-art learning-based methods in both solution quality and generalization ability.