A recent study on the classical capacitated vehicle routing problem (CVRP) introduced an adaptive version of the widely used iterated local search paradigm, hybridized with a path-relinking (PR) strategy. The solution method, called adaptive iterated local search (AILS)-PR, outperformed existing meta-heuristics for the CVRP on benchmark instances. However, tests on large-scale instances suggest that PR is too slow, making AILS-PR less advantageous in this case. To overcome this challenge, this paper presents an AILS combined with mechanisms to handle large CVRP instances, called AILS-II. The computational cost of this implementation is reduced, whereas the algorithm also searches the solution space more efficiently. AILS-II is very competitive on smaller instances, outperforming the other methods from the literature with respect to the average gap to the best-known solutions. Moreover, AILS-II consistently outperforms the state of the art on larger instances with up to 30,000 vertices. History: Accepted by Ted Ralphs, Area Editor for Software Tools. This paper has been accepted for the INFORMS Journal on Computing Special Issue on Software Tools for Vehicle Routing. Funding: This work was supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo [Grants 2013/07375-0, 2019/22067-6, and 2022/05803-3] and the Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 309385/2021-0 and 403735/2021-1]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0106 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0106 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

In this paper, we propose a new semi-supervised growing neural gas (GNG) model, named Consensus-Based Semi-Supervised GNG, or CSSGNG, in which both labeled and unlabeled data are used to train the network. In contrast to former adaptations of the GNG to semi-supervised classification, such as the SSGNG and OSSGNG models, the CSSGNG does not assign a single scalar label value to each neuron. Instead of the scalar, a vector containing the representativeness level of every class is associated with each neuron. Moreover, to propagate the labels among the neurons the CSSGNG employs a consensus approach. Computer experiments show that our model on average can deliver better classification results in comparison to the SSGNG and OSSGNG models.

Network motif algorithms have been a topic of research mainly after the 2002-seminal paper from Milo et al. [1], which provided motifs as a way to uncover the basic building blocks of most networks. Motifs have been mainly applied in Bioinformatics, regarding gene regulation networks. Motif detection is based on induced subgraph counting. This paper proposes an algorithm to count subgraphs of size k + 2 based on the set of induced subgraphs of size k. The general technique was applied to detect 3, 4 and 5-sized motifs in directed graphs. Such algorithms have time complexity O(a(G)m), O(m(2)) and O(nm(2)), respectively, where a(G) is the arboricity of G(V, E). The computational experiments in public data sets show that the proposed technique was one order of magnitude faster than Kavosh and FANMOD. When compared to NetMODE, acc-Motif had a slightly improved performance.