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Consider the following integer program (general covering):

$\min c \cdot x$ subject to

$Ax \ge b$,

where all entries in $A, b, c$ are nonnegative and $x$ is required to be nonnegative and integral.

In Vazirani, it is an exercise to show that an $O( \log n )$-approximation algorithm exists, where $A$ is an $n$ by $m$ matrix. Could someone give me a reference on how to do this?

In particular, I am interested in the case when $x$ is required to be binary, and in the use of linear programming to approximate.

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    $\begingroup$ If you really want the variables to be binary, it's a moderately challenging exercise. Here's a paper about it dl.acm.org/citation.cfm?id=1119632 that gives one method. It can be simplified a bit. $\endgroup$ – Neal Young Apr 12 '13 at 5:28

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