通过行列式的精确覆盖

摘要：

给定一个n个顶点的k-一致超图，将其划分为k个相等的部分，使得每个超边包含来自每个部分的一个顶点，k维匹配问题询问是否存在一个覆盖所有顶点的不相交超边集合。我们证明了它可以用一个随机多项式空间算法在O*（2^（n（k-2）/k））时间内求解。O*（）表示法隐藏了n和k中的多项式因子。当我们去掉划分约束，允许任意基数为k的超边时，我们得到了精确的k-集覆盖问题。我们证明了它可以用一个随机多项式空间算法在O*（c_k^n）内求解，其中c_3=1.496，c_4=1.642，c_5=1.721，并给出了k越大时的一般界.这两个结果都大大改进了之前解决这些问题的最佳算法，尤其是对于小k，并且遵循了新的观察结果，即Lovasz通过决定因素进行完美匹配检测（1979）承认嵌入到最近提出的集合包含-排除计数方案中覆盖，尽管它无法计算完美匹配。

英文摘要：

Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such that every hyperedge includes one vertex from each part, the k-dimensional matching problem asks whether there is a disjoint collection of the hyperedges which covers all vertices. We show it can be solved by a randomized polynomial space algorithm in time O*(2^(n(k-2)/k)). The O*() notation hides factors polynomial in n and k. When we drop the partition constraint and permit arbitrary hyperedges of cardinality k, we obtain the exact cover by k-sets problem. We show it can be solved by a randomized polynomial space algorithm in time O*(c_k^n), where c_3=1.496, c_4=1.642, c_5=1.721, and provide a general bound for larger k. Both results substantially improve on the previous best algorithms for these problems, especially for small k, and follow from the new observation that Lovasz' perfect matching detection via determinants (1979) admits an embedding in the recently proposed inclusion-exclusion counting scheme for set covers, despite its inability to count the perfect matchings.

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