Which is the approximation class of Minimum 0-1 integer programming if only non-negative integers are allowed? I'm wondering which is the approximation class of Minimum 0-1 Integer Programming if only non-negative integers are used. That is:


minimize $c^T \cdot x$, 
with $x\in \{0,1\}^n$ and $c\in (\mathbb{Z}^*)^n$, 
subject to $Ax \geq b$, where $A\in (\mathbb{Z}^*)^{m * n}$ and $b\in (\mathbb{Z}^*)^m$.


where $c^T$ is the transpose of vector $c$ and $\mathbb{Z}^*$ is the set of non-negative integers. Alternatively, $\mathbb{Z}^*$ could be replaced by the set of non-negative rationals, and we could force $b=(1,...,1)$ (it would be the same problem up to normalization).

If negative integers were allowed in $A$ and $b$, then this problem would be Minimum 0-1 Integer Programming, which is NPO-complete.

If $x$ were a vector of integers, rather than a vector of bits, then it would be Cover Integer Program (CIP), which can be approximated logarithmically according to "Improved approximations of packing and covering problems" by A. Srinivasan (1995).

There are many other known related problems with different variations, e.g. all elements in $A$ must be bits, $\geq$ is $\leq$ and min is max, there is a single inequation (i.e. $m = 1$), etc. I had some hope when I found this: https://www.nada.kth.se/~viggo/wwwcompendium/node199.html#7105, comments section, last sentence. However, $A$ is a matrix of bits in that work.

It is trivial to build an S-reduction from Minimum Set Cover, so at least it is Log-APX-hard. But at a first glance it looks too "simple" for Poly-APX- or Exp-APX-hardness, I don't know... Has this problem ever been studied?

EDIT: In the last two weeks I have developed an approximation to the problem (not based on linear programming) and, if my maths are right, it is logarithmic. Since it looks like this variant of  Minimum 0-1 Integer Programming has not been studied yet in the literature, it could be publishable (?). I promise I'll get back here and report it, as soon as some reviewers check out its correctness (also if it's wrong, so that nobody else wastes his time trying the same!). (And, of course, if this problem has already been tackled in the literature, I would really appreciate it if you let me know!!)

EDIT2: As Chandra's answer says below, Kolloiopoulos and Young gave a logarithmic approximation to a generalization of this problem. They define $x$ as a vector of non-negative integers but, simultaneously, include constraints of the form $x \leq d$. By setting $d=(1,...,1)$, $x$ must be a vector of bits.
Kolloiopoulos and Young give an $O(\log m)$ approximation for general covering integer programs. See the paper below.
http://www.sciencedirect.com/science/article/pii/S0022000005000656