Graph layout problems refer to a family of optimization problems with relevant applications in VLSI design, information retrieval, numerical analysis, computational biology, or graph theory. In this paper, we focus on a variant where the graph is mapped over a specific structure referred to as book, which consists of a spine and a number of pages. Vertices of the graph are allocated in the spine, which is usually represented as a line, and edges are assigned to pages of the book, which are represented as half-planes that have the spine as its boundary. The experience shows that one of the main quality desired for layout of graphs is the minimization of edge crossing. The $K$-page crossing number minimization problem (KPMP) aims to reduce as much as possible the number of crossings between edges belonging to the same page. We propose the application of the VNS metaheuristic to efficiently solve the KPMP. Experiments show a remarkable improvement over the state-of-the-art procedures for a large set of instances, leading the proposed VNS algorithm as a promising strategy to solve this family of book drawing problems.

图形布局问题是指一系列优化问题，在VLSI设计，信息检索，数值分析，计算生物学或图形理论中具有相关应用。在本文中，我们重点介绍一种变体，其中图形映射到称为书的特定结构上，该结构由书脊和许多页组成。图的顶点分配在书脊中，通常用一条线表示，而边线则分配给该书的各页，这些书页用以书脊为边界的半平面表示。经验表明，图形布局所需的主要质量之一是最小化边缘交叉。的$ķ$页穿越次数最小化问题（KPMP）旨在尽可能减少属于同一页面的边缘之间的穿越次数。我们提出了应用VNS元启发式算法有效解决KPMP的问题。实验表明，对于大量实例而言，与现有技术相比，该方法已取得了显着进步，从而使提出的VNS算法成为解决该系列图书绘制问题的有前途的策略。