Even, Itai, and Shamir (1976) proved simple two‐commodity integral flow is NP‐complete both in the directed and undirected cases. In particular, the directed case was shown to be NP‐complete even if one demand is unitary, which was improved by Fortune, Hopcroft and Wyllie (1980) who proved the problem is still NP‐complete if both demands are unitary. The undirected case, on the other hand, was proved by Robertson and Seymour (1995) to be polynomial‐time solvable if both demands are constant. Nevertheless, the complexity of the undirected case with exactly one constant demand has remained unknown. We close this 40‐year complexity gap, by showing the undirected case is NP‐complete even if exactly one demand is unitary. As a by product, we obtain the NP‐completeness of determining whether a graph contains 1 + d pairwise vertex‐disjoint paths, such that one path is between a given pair of vertices and d paths are between a second given pair of vertices. Additionally, we investigate the complexity of another related network design problem called strict terminal connection.

中文翻译：

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关于无方向的两商品积分流，不相交的路径和严格的终端连接问题

甚至，Itai和Shamir（1976）证明，在有向和无向情况下，简单的两商品积分流都是NP完全的。尤其是，即使一个需求是单一的，定向案例也被证明是NP完全的，Fortune，Hopcroft和Wyllie（1980）对此进行了改进，他们证明了如果两个需求都是单一的，问题仍然是NP完全的。另一方面，罗伯逊和西摩（Robertson and Seymour，1995）证明了无方向情况，如果两个要求都恒定，则是多项式时间可解的。然而，只有一个恒定需求的无定向案件的复杂性仍然未知。通过证明无方向案例是NP完全的，即使恰好一个需求是单一的，我们也弥合了40年的复杂性差距。作为副产品，我们获得确定图是否包含1 +  d的NP完整性。逐对顶点不相交的路径，例如，一条路径在给定的一对顶点之间，而d路径在第二对给定的顶点之间。此外，我们调查了另一个相关网络设计问题（称为严格终端连接）的复杂性。