Technical Report CS0913

Title: Constant Absolute Error Approximation Algorithm for the ``Safe Deposit Boxes'' Problem
Authors: Yefim Dinitz
PDF - RevisedCS0913.revised.pdf
Abstract: Suppose there are $n$ safe deposit boxes, each containing known amounts of $m$ currencies, and there is a certain need for these currencies. The problem is how to open the minimal number of boxes in order to collect at least the prescribed amount of each currency. This problem is proved to be NP-hard. For the case $m=2$, an algorithm with an absolute error of at most 1 is suggested, with complexity $O(n^2 \log n)$. For general $m$, we observe (together with A.\ V.\ Karzanov) that the ceiling rounding of an extreme-point solution for the corresponding integral linear program is an approximation with an absolute error of at most $m-1$; for $m$ fixed, $O(n^{m+1})$ time being sufficient to obtain such a solution.
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