G.1 Monoids Monoids are one of the simplest algebraic structure.

There are two more things to verify.

THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICA TION. Abstract--The capacitated location-routing problems (CLRPs) are classical problems in combinatorial optimization, which require simultaneously making location and routing decisions. In CLRPs, the complex constraints and the intricate relationships between various decisions make the problem challenging to solve. With the emergence of deep reinforcement learning (DRL), it has been extensively applied to address the vehicle routing problem and its variants, while the research related to CLRPs still needs to be explored. In this paper, we propose the DRL with heterogeneous query (DRLHQ) to solve CLRP and open CLRP (OCLRP), respectively. We are the first to propose an end-to-end learning approach for CLRPs, following the encoder-decoder structure. In particular, we reformulate the CLRPs as a markov decision process tailored to various decisions, a general modeling framework that can be adapted to other DRL-based methods. T o better handle the interdependency across location and routing decisions, we also introduce a novel heterogeneous querying attention mechanism designed to adapt dynamically to various decision-making stages. Experimental results on both synthetic and benchmark datasets demonstrate superior solution quality and better generalization performance of our proposed approach over representative traditional and DRL-based baselines in solving both CLRP and OCLRP . HE facility location problem (FLP) and vehicle routing problem (VRP) are two critical combinatorial optimization problems (COPs) in transportation and logistics, which are traditionally addressed sequentially. However, planning the routes after facility location may lead to suboptimal solutions due to the interdependencies across various decisions [1], [2]. Therefore, the capacitated location-routing problems (CLRPs) [3] are proposed to simultaneously make location and routing decisions. The CLRPs are one of the most classical topics in the community of operations research and have extensive applications such as supply-chain management [4], emergency management [5], and disaster relief [6]. This work was supported by the National Key Research and Development Program of China No.2022ZD0119703; in part by the National Natural Science Foundations of China (NSFC) under Grant 62273044 and 62022015; in part by the National Natural Science Foundation of China National Science Fund for Distinguished Y oung Scholars under Grant 62025301; in part by the National Natural Science Foundation of China Basic Science Center Program under Grant 62088101. In CLRPs, depots and vehicles are subject to the maximum capacity constraints, and the depots are considered heterogeneous due to distinct capacities and opening costs. Meanwhile, we also study the open CLRP (OCLRP) [7], a variant of CLRP, by considering open-ended routes.

The Edge Assembly Crossover (EAX) algorithm is the state-of-the-art heuristic for solving the Traveling Salesperson Problem (TSP). It regularly outperforms other methods, such as the Lin-Kernighan-Helsgaun heuristic (LKH), across diverse sets of TSP instances. Essentially, EAX employs a two-stage mechanism that focuses on improving the current solutions, first, at the local and, subsequently, at the global level. Although the second phase of the algorithm has been thoroughly studied, configured, and refined in the past, in particular, its first stage has hardly been examined. In this paper, we thus focus on the first stage of EAX and introduce a novel method that quickly verifies whether the AB-cycles, generated during its internal optimization procedure, yield valid tours -- or whether they need to be repaired. Knowledge of the latter is also particularly relevant before applying other powerful crossover operators such as the Generalized Partition Crossover (GPX). Based on our insights, we propose and evaluate several improved versions of EAX. According to our benchmark study across 10 000 different TSP instances, the most promising of our proposed EAX variants demonstrates improved computational efficiency and solution quality on previously rather difficult instances compared to the current state-of-the-art EAX algorithm.

While behavior cloning is straightforward, it requires expensive expert solutions, and policy gradient methods are often computationally demanding and complex to fine-tune. In this work, we bridge the two and simplify the training process by sampling multiple solutions for random instances using the current model in each epoch and then selecting the best solution as an expert trajectory for supervised imitation learning. To achieve progressively improving solutions with minimal sampling, we introduce a method that combines round-wise Stochastic Beam Search with an update strategy derived from a provable policy improvement. This strategy refines the policy between rounds by utilizing the advantage of the sampled sequences with almost no computational overhead. We evaluate our approach on the Traveling Salesman Problem and the Capacitated Vehicle Routing Problem. The models trained with our method achieve comparable performance and generalization to those trained with expert data. Additionally, we apply our method to the Job Shop Scheduling Problem using a transformer-based architecture and outperform existing state-of-the-art methods by a wide margin.

The convex hull cheapest insertion heuristic is known to generate good solutions to the Euclidean Traveling Salesperson Problem. This paper presents an adaptation of this heuristic to the non-Euclidean version of the problem and further extends it to the problem with precedence constraints, also known as the Sequential Ordering Problem. To test the proposed algorithm, the well-known TSPLIB benchmark data-set is modified in a replicable manner to create non-Euclidean instances and precedence constraints. The proposed algorithm is shown to outperform the commonly used Nearest Neighbor algorithm in 97% of the cases that do not have precedence constraints. When precedence constraints exist such that the child nodes are centrally located, the algorithm again outperforms the Nearest Neighbor algorithm in 98% of the studied instances. Considering all spatial layouts of precedence constraints, the algorithm outperforms the Nearest Neighbor heuristic 68% of the time.

This paper further defines the class of fragment constructive heuristics used to compute feasible solutions for the Traveling Salesman Problem into arc-greedy and node-greedy subclasses. Since these subclasses of heuristics can create subtours, two known methodologies for subtour elimination on symmetric instances are reviewed and are expanded to cover asymmetric problem instances. This paper introduces a third novel methodology, the Greedy Tracker, and compares it to both known methodologies. Computational results are generated across multiple symmetric and asymmetric instances. The results demonstrate the Greedy Tracker is the fastest method for preventing subtours for instances below 400 nodes. A distinction between fragment constructive heuristics and the subtour elimination methodology used to ensure the feasibility of resulting solutions enables the introduction of a new node-greedy fragment heuristic called Ordered Greedy.