组合拓扑学中的固定参数易处理算法
Fixed parameter tractable algorithms in combinatorial topology
摘要：
为了枚举具有给定属性的3-流形三角剖分，通常从一组潜在的面配对图（也称为双1-骨架）开始，然后尝试使用指数时间枚举将每个图充实成完整的三角剖分。然而，渐近大多数图不导致任何3-流形三角剖分，这导致显着的“浪费时间”的拓扑枚举算法。在这里，我们给出了一个新的算法，以确定是否一个给定的脸配对图支持任何3-流形三角剖分，并显示这是固定的参数在树宽的图易处理。我们将这一结果扩展到一个“元定理”，通过定义一个广泛的三角剖分的性质，每个与相应的固定参数听话的存在算法。我们明确地实现这个算法在最通用的设置，我们确定在实践中被认为是减轻大常数，经常发生在参数化的复杂性，突出了我们的技术的实用性。
英文摘要：
To enumerate 3-manifold triangulations with a given property, one typically begins with a set of potential face pairing graphs (also known as dual 1-skeletons), and then attempts to flesh each graph out into full triangulations using an exponential-time enumeration. However, asymptotically most graphs do not result in any 3-manifold triangulation, which leads to significant "wasted time" in topological enumeration algorithms. Here we give a new algorithm to determine whether a given face pairing graph supports any 3-manifold triangulation, and show this to be fixed parameter tractable in the treewidth of the graph. We extend this result to a "meta-theorem" by defining a broad class of properties of triangulations, each with a corresponding fixed parameter tractable existence algorithm. We explicitly implement this algorithm in the most generic setting, and we identify heuristics that in practice are seen to mitigate the large constants that so often occur in parameterised complexity, highlighting the practicality of our techniques.
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