A Fun Optimization Problem

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I spent the last several hours trying to come up with an efficient algorithm to the following problem:

Problem: Suppose that we have a sequence of $l$ pairs of non-negative numbers $(a_1,b_1),\ldots,(a_l,b_l)$ such that $\sum_{i=1}^l a_i \leq A$ and $\sum_{i=1}^l b_i \leq B$. Devise an efficient algorithm to find the $k$ pairs $(a_{i_1},b_{i_1}),\ldots,(a_{i_k},b_{i_k})$ that maximize

$\left[\sum_{r=1}^k a_{i_r}\log(a_{i_r}/b_{i_r})\right] + \left[A-\sum_{r=1}^k a_{i_r}\right]\log\left(\frac{A-\sum_{r=1}^k a_{i_r}}{B-\sum_{r=1}^k b_{i_r}}\right).$

Commentary: I don’t have a fully satisfactory solution to this yet, although I do think I can find an algorithm that runs in $O\left(\frac{l \log(l)}{\epsilon}\right)$ time and finds $2k$ pairs that do at least $1-\epsilon$ as well as the best set of $k$ pairs. It’s possible I need to assume something like $\sum_{i=1}^l a_i \leq A/2$ instead of just $A$ (and similarly for the $b_i$), although I’m happy to make that assumption.

While attempting to solve this problem, I’ve managed to utilize a pretty large subset of my bag of tricks for optimization problems, so I think working on it is pretty worthwhile intellectually. It also happens to be important to my research, so if anyone comes up with a good algorithm I’d be interested to know.

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