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Given: Set $N = \{0, \ldots, n-1\}, k \in \mathbb{N}, d_{ij} \geq 0$ with $d_{ii} = 0$.

Task: Find subset $C \subseteq N, |C| \leq k$ that minimizes $\max_{i \in N} \min_{j \in C} d_{ij}$.

Idea: I define $x_i$ with $x_i = 1$ if $i \in C$ and zero otherwise, constrained by $\sum_{i \in N} x_i \leq k$.

Now I want to define $w_i$, constrained with $w_i \geq d_{ij}$ for one arbitrary $j$ with $x_j = 1$. Here lies my biggest problem. I can write these things in python but not as a linear program. No matter what I try, I lose linearity or something else in my formulation.

If I can do this step, I could define $y$, constrained by $y \geq w_i$ for every $i \in N$ and minimize $y$.

I think I have a major flaw in my idea, but after thinking a lot about it, I can't come up with a different approach, probably because this is the first LP I have to write.

Does anyone have any idea as to how I could formulate my problem as an LP?

Yuval Filmus
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Niah
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  • Why do you want to solve this using an LP? – Yuval Filmus Nov 14 '21 at 13:10
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    Using binary search on the weights, you can reduce the general version to the special case in which the weights are ${0,1}$, and so define a digraph. Can you solve this version? – Yuval Filmus Nov 14 '21 at 13:18
  • To elaborate a bit on the above, you use binary search to guess the answer, i.e. you ask "is it true that the answer is at most $a$". Such questions, as explained above, are easy to answer, and they suffice to use binary search. – Dmitry Nov 15 '21 at 07:24
  • Also, when you see a problem of form $\max \min \to \min$, it's an indicator that you can't use LP. LP problems are inherently convex, while such problems are usually nonconvex. – Dmitry Nov 15 '21 at 07:26
  • This looks like a coreset selection problem, for which you could refer to Sener and Savarese (2018). – SnzFor16Min Mar 26 '24 at 02:28

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