In the assignment problem, one tries to find $f$ such that the cost function $$ \sum_{a\in A} C(a,f(a)) $$ is minimized. Here $f$ is a bijection between sets $A$ and $B$ of equal finite cardinality, and $C$ is a cost function $A\times B \to \mathbb R$.

My question: is there a name for the problem to minimize $$ \max_{a\in A} \, C(a,f(a)) $$ over bijections $f$ as above?

Thanks in advance for any replies or references.

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    $\begingroup$ (1) I think that I have seen this variant on this website, but I cannot locate it. (2) Because the answer is one of the values of C, you can use binary search. But maybe there is a faster algorithm. $\endgroup$ – Tsuyoshi Ito Feb 21 '12 at 3:25

In the geometric setting, where $C(x,y) = \|x - y \|$, this formulation is called the bottleneck matching problem. It's possible that this is the generic term for it (I've seen this formulation used in the Kleinberg-Tardos algorithms book for MSTs).

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  • 7
    $\begingroup$ It's also called the bottleneck assignment problem $\endgroup$ – Ryan Williams Feb 21 '12 at 4:37

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