The balanced incomplete block design (BIBD) problem is a difficult combinatorial problem with a large number of symmetries, which add complexity to its resolution. In this paper, we propose a dual (integer) problem representation that serves as an alternative to the classical binary formulation of the problem. We attack this problem incrementally: firstly, we propose basic algorithms (i.e. local search techniques and genetic algorithms) intended to work separately on the two different search spaces (i.e. binary and integer); secondly, we propose two hybrid schemes: an integrative approach (i.e. a memetic algorithm) and a collaborative model in which the previous methods work in parallel, occasionally exchanging information. Three distinct two-dimensional structures are proposed as communication topology among the algorithms involved in the collaborative model, as well as a number of migration and acceptance criteria for sending and receiving data. An empirical analysis comparing a large number of instances of our schemes (with algorithms possibly working on different search spaces and with/without symmetry breaking methods) shows that some of these algorithms can be considered the state of the art of the metaheuristic methods applied to finding BIBDs. Moreover, our cooperative proposal is a general scheme from which distinct algorithmic variants can be instantiated to handle symmetrical optimisation problems. For this reason, we have also analysed its key parameters, thereby providing general guidelines for the design of efficient/robust cooperative algorithms devised from our proposal.

翻译：平衡不完全区组设计（BIBD）问题是一个具有大量对称性的困难组合问题，这些对称性增加了其求解复杂度。本文提出了一种对偶（整数）问题表示法，作为该问题经典二元表述的替代方案。我们采用渐进式策略处理此问题：首先，我们提出了旨在分别作用于两个不同搜索空间（即二元空间和整数空间）的基础算法（如局部搜索技术和遗传算法）；其次，我们提出了两种混合方案：一种整合式方法（即模因算法）和一种协同模型，其中前述方法并行运行并偶尔交换信息。我们为协同模型中涉及的算法提出了三种不同的二维结构作为通信拓扑，并制定了若干用于发送和接收数据的迁移与接受准则。通过对我们方案的大量实例（包含可能在不同搜索空间运行及使用/未使用对称性破缺方法的算法）进行实证分析，结果表明其中某些算法可被视为应用于寻找BIBD问题的元启发式方法的先进代表。此外，我们的协同方案是一个通用框架，可从中实例化出不同的算法变体来处理对称优化问题。为此，我们还分析了其关键参数，从而为基于本方案设计高效/稳健的协同算法提供了通用指导原则。