Difficult, in particular NP-complete, optimization problems are traditionally solved approximately using search heuristics. These are usually slowed down by the rugged landscapes encountered, because local minima arrest the search process. Cover-encoding maps were devised to circumvent this problem by transforming the original landscape to one that is free of local minima and enriched in near-optimal solutions. By definition, these involve the mapping of the original (larger) search space into smaller subspaces, by processes that typically amount to a form of coarse-graining. In this paper, we explore the details of this coarse-graining using formal arguments, as well as concrete examples of cover-encoding maps, that are investigated analytically as well as computationally. Our results strongly suggest that the coarse-graining involved in cover-encoding maps bears a strong resemblance to that encountered in renormalisation group schemes. Given the apparently disparate nature of these two formalisms, these strong similarities are rather startling, and suggest deep mathematical underpinnings that await further exploration.

翻译：困难（特别是NP完全）的优化问题传统上通过搜索启发式方法进行近似求解。这些方法通常因遇到崎岖的景观而减慢速度，因为局部极小值会阻碍搜索过程。覆盖编码映射旨在通过将原始景观转换为无局部极小值且富含近优解的地形来规避这一问题。根据定义，这些映射涉及将原始（较大）搜索空间映射到较小子空间的过程，通常相当于一种粗粒化操作。本文通过形式化论证以及解析与计算研究的具体覆盖编码映射实例，深入探讨了这种粗粒化的细节。我们的结果强烈表明，覆盖编码映射中的粗粒化过程与重正化群方案中遇到的粗粒化过程高度相似。考虑到这两种形式化方法表面上的迥异性质，这些显著的相似性相当令人惊讶，并暗示了有待进一步探索的深层数学基础。