We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. Consequently, we get an algorithm for enumerating all the proper tree decompositions, in incremental polynomial time, where "proper" means that the tree decomposition cannot be improved by removing or splitting a bag. The algorithm can incorporate any method for (ordinary, single result) triangulation or tree decomposition, and can serve as an anytime algorithm to improve such a method. We describe an extensive experimental study of an implementation on real data from different fields. Our experiments show that the algorithm improves upon central quality measures over the underlying tree decompositions, and is able to produce a large number of high-quality decompositions.

翻译：我们提出了一种算法，能够在增量多项式时间内枚举图的所有最小三角剖分。由此，我们进一步得到一种算法，可在增量多项式时间内枚举所有“恰当”的树分解，其中“恰当”意为树分解无法通过移除或拆分一个袋子而得到改进。该算法可兼容任何（单结果）三角剖分或树分解方法，并可作为任意时刻算法来改进此类方法。我们报告了在来自不同领域的真实数据上对算法实现进行的广泛实验研究。实验表明，该算法在底层树分解的核心质量指标上优于原有方法，并能生成大量高质量分解。