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I have been trying to find the algorithmic complexity of a problem that I have. I am almost sure it is either NP-hard or NP-complete but I cannot find any proof. Recently, I found that my problem can be something similar to a special instance of the Maximum Capacity Representatives problem, which is NP-complete. However, the objective function to optimize in my case is different than the one in the MCR problem.

The problem that I am trying to solve is the following:

INSTANCE: Disjoint sets $S_1, \ldots, S_m$ and, for any $i \neq j$, $x \in S_i$, and $y \in S_j$, a nonnegative capacity $c(x,y)$.

SOLUTION: A system of representatives $T$, i.e., a set $T$ such that, for any $i$, $\vert T \cap S_i\vert=1$.

MEASURE: $\min \{c(x,y): x,y \in T \}$.

And my goal is to maximize $\min \{c(x,y): x,y \in T \}$.

Do you know any way to determine the complexity of my problem? Is there any well known problem in the literature that can be reduced to this one?,

Thanks in advance.

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I'm assuming that the goal of your optimization problem is to maximize $\min \{ c(x,y) : x,y \in T \}$. The decision problem is then:

Given a system $S_1,\ldots,S_m$ of disjoint sets, a cost function $c\colon S \times S \to \mathbb{R}_+$ (where $S = S_1 \cup \cdots \cup S_m$) and a number $\gamma \in \mathbb{R}_+$, decide whether there is a choice of representatives $t_i \in S_i$ such that $c(t_i,t_j) \geq \gamma$ for all $i \neq j$.

This problem is NP-hard (and so NP-complete), by reduction from max clique. Given a graph $G = (V,E)$ and a number $m$, let $S_i = \{i\} \times V$, define $c((i,v),(j,w))$ to be 1 if $(v,w) \in E$ and 0 otherwise, and let $\gamma = 1$. The graph $G$ contains an $m$-clique if and only if the answer to your decision problem is affirmative.

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  • $\begingroup$ Thanks, but I am still confused in one part. If I understand well, for this proof I don't care about the real values of the costs, I map them to be 1 or 0. However, I don't understand how to decide whether they are mapped to 1 or 0, it depends on the existence of the edge $(v,w)$, but when does $(v,w)$ exists in $E$? $\endgroup$ – Guillermo Jun 25 '17 at 5:03
  • $\begingroup$ The input to the max clique problem is a graph and an integer. A graph is a list of vertices and edges. Given two vertices, you can use the list to decide whether two vertices are connected. $\endgroup$ – Yuval Filmus Jun 25 '17 at 5:04
  • $\begingroup$ Is the value $\gamma=1$ due to the mapping of the costs, right?, I am sorry if I am having trouble with some details, this is the first time that I see the max clique problem (I am reading about it). $\endgroup$ – Guillermo Jun 25 '17 at 5:19
  • $\begingroup$ You can choose any value in $(0,1]$. My choice was arbitrary. $\endgroup$ – Yuval Filmus Jun 25 '17 at 5:20
  • $\begingroup$ Ok, Thank you so much Yuval, you were very helpful. I think I can get the rest of the details by myself. $\endgroup$ – Guillermo Jun 25 '17 at 5:24

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