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This chapter delves into the emerging field of neuro-symbolic query optimization for knowledge graphs (KGs), presenting a comprehensive exploration of how neural and symbolic techniques can be integrated to enhance query processing. Traditional query optimizers in knowledge graphs rely heavily on symbolic methods, utilizing dataset summaries, statistics, and cost models to select efficient execution plans. However, these approaches often suffer from misestimations and inaccuracies, particularly when dealing with complex queries or large-scale datasets. Recent advancements have introduced neural models, which capture non-linear aspects of query optimization, offering promising alternatives to purely symbolic methods. In this chapter, we introduce neuro-symbolic query optimizers, a novel approach that combines the strengths of symbolic reasoning with the adaptability of neural computation. We discuss the architecture of these hybrid systems, highlighting the interplay between neural and symbolic components to improve the optimizer's ability to navigate the search space and produce efficient execution plans. Additionally, the chapter reviews existing neural components tailored for optimizing queries over knowledge graphs and examines the limitations and challenges in deploying neuro-symbolic query optimizers in real-world environments.

Cardinality Estimation over Knowledge Graphs (KG) is crucial for query optimization, yet remains a challenging task due to the semi-structured nature and complex correlations of typical Knowledge Graphs. In this work, we propose GNCE, a novel approach that leverages knowledge graph embeddings and Graph Neural Networks (GNN) to accurately predict the cardinality of conjunctive queries. GNCE first creates semantically meaningful embeddings for all entities in the KG, which are then integrated into the given query, which is processed by a GNN to estimate the cardinality of the query. We evaluate GNCE on several KGs in terms of q-Error and demonstrate that it outperforms state-of-the-art approaches based on sampling, summaries, and (machine) learning in terms of estimation accuracy while also having lower execution time and less parameters. Additionally, we show that GNCE can inductively generalise to unseen entities, making it suitable for use in dynamic query processing scenarios. Our proposed approach has the potential to significantly improve query optimization and related applications that rely on accurate cardinality estimates of conjunctive queries.