[英]Is this assignment problem with constrains NP-hard?
It is a many-to-one assignment problem with N tasks and M people.这是一个有 N 个任务和 M 个人的多对一分配问题。
Each person can get multiple tasks, while each task can be assigned to only one person.每个人可以获得多个任务,而每个任务只能分配给一个人。 We can earn a profit Pij if the task i is assigned to person j.
如果将任务 i 分配给人员 j,我们可以获得利润 Pij。
If T1, T2,..., Tm is a partition of the tasks, and n1, n2,..., nm are m positive integers.如果 T1, T2,..., Tm 是任务的一个分区,并且 n1, n2,..., nm 是 m 个正整数。 I want the optimum assignment such that the number of people assigned to any task in Ti must be less or equal to ni
我想要最佳分配,使得分配给 Ti 中任何任务的人数必须小于或等于 ni
If I understand your question correctly, this is a special case of the minimum-cost flow problem on a graph with three layers (in addition to a source and a sink layer).如果我正确理解您的问题,这是具有三层(除了源和汇层)图上的最小成本流问题的特例。
We haven't specified a demand, but we could simply try all possible demands between 0 and M and still be in P , so showing that it's not NP -hard is equivalent to showing that P ≠ NP .我们没有指定需求,但我们可以简单地尝试 0 和M之间的所有可能需求,并且仍然在P中,因此表明它不是NP难的就等于表明P ≠ NP 。
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