Given matrix $A \in \{0,1\}^{m \times n}$ and vector $b \in (\mathbb{Z^+})^m$, where $\mathbb{Z^+}$ is the set of positive integers,
$$\begin{array}{ll} \text{maximize} & c^T x\\ \text{subject to} & Ax \leq b\\ & x \geq 0\\ & x \in \{0,1\}^n\end{array}$$
Notice the biggest difference from normal $0-1$ integer programming is that $A \in \{0,1\}^{m \times n}$ and $b \in (\mathbb{Z^+})^m$. Is there anything special about such integer programs? Is there an algorithm to solve them in polynomial time?