Abstract A connectivity function on a set E is a function λ : 2 E → R such that λ ( 0 ) = 0 , that λ ( X ) = λ ( E − X ) for all X ⊆ E , and that λ ( X ∩ Y ) + λ ( X ∪ Y ) ≤ λ ( X ) + λ ( Y ) for all X , Y ⊆ E . Graphs, matroids and, more generally, polymatroids have associated connectivity functions. In this paper we give a method for identifying when a connectivity function comes from a graph. This method uses no more than a polynomial number of evaluations of the connectivity function. In contrast, we show that the problem of identifying when a connectivity function comes from a matroid cannot be solved in polynomial time. We also show that the problem of identifying when a connectivity function is not that of a matroid cannot be solved in polynomial time.

中文翻译：

---

识别图形和拟阵连通性函数

摘要 集合 E 上的连通函数是一个函数 λ : 2 E → R 使得 λ ( 0 ) = 0 ，对于所有 X ⊆ E λ ( X ) = λ ( E − X ) ，并且 λ ( X ∩ Y ) + λ ( X ∪ Y ) ≤ λ ( X ) + λ ( Y ) 对于所有 X , Y ⊆ E 。图、拟阵以及更一般的多拟阵具有相关联的连通性函数。在本文中，我们给出了一种识别连通性函数何时来自图的方法。该方法仅使用多项式数的连接函数评估。相比之下，我们表明无法在多项式时间内解决确定连接函数何时来自拟阵的问题。我们还表明，无法在多项式时间内解决识别连接函数何时不是拟阵的问题。