平面图中k个不匹配点的匹配计数
Counting matchings with k unmatched vertices in planar graphs
摘要：
我们考虑平面图中匹配的计数问题。虽然平面图中的完美匹配可以通过经典的多项式时间算法来计数，但已知计数所有匹配（可能包含不匹配的顶点，也称为缺陷）的问题在平面图上是#P-完全的。为了在计算匹配的困难情况和计算完美匹配的简单情况之间进行插补，我们研究了平面图G中具有恰好k个不匹配顶点的匹配的参数化计算问题，关于输入G和k.这个设置在统计物理学中有一个自然的解释，它是计算k顶点图（可以通过删除最多k个顶点而变成平面的图）中完美匹配的特殊情况。从最近关于k-顶点图上完美匹配计数的一个#W[1]-困难证明出发，得到了平面图中k个顶点不匹配的匹配计数是#W[1]-困难的.与此相反，给定一个平面图G的s个区别的面，有一个$O(2^s \cdot n^3)$时间算法，用于计数那些匹配与k个不匹配的顶点，使所有不匹配的顶点位于区别的面。这意味着一个$f(k,s)\cdot n^{O(1)}$时间算法计算完美匹配的k顶点图的顶点邻域覆盖的s面。
英文摘要：
We consider the problem of counting matchings in planar graphs. While perfect matchings in planar graphs can be counted by a classical polynomial-time algorithm, the problem of counting all matchings (possibly containing unmatched vertices, also known as defects) is known to be #P-complete on planar graphs. To interpolate between the hard case of counting matchings and the easy case of counting perfect matchings, we study the parameterized problem of counting matchings with exactly k unmatched vertices in a planar graph G, on input G and k. This setting has a natural interpretation in statistical physics, and it is a special case of counting perfect matchings in k-apex graphs (graphs that can be turned planar by removing at most k vertices). Starting from a recent #W[1]-hardness proof for counting perfect matchings on k-apex graphs, we obtain that counting matchings with k unmatched vertices in planar graphs is #W[1]-hard. In contrast, given a plane graph G with s distinguished faces, there is an $O(2^s \cdot n^3)$ time algorithm for counting those matchings with k unmatched vertices such that all unmatched vertices lie on the distinguished faces. This implies an $f(k,s)\cdot n^{O(1)}$ time algorithm for counting perfect matchings in k-apex graphs whose apex neighborhood is covered by s faces.
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