  This paper presents an anlysis of the NP-hard minimization problem $\min
\{\|b - Ax\|_2: \ x \in [0,1]^n, | \text{supp}(x) | \leq \sigma\}$, where
$\text{supp}(x) = \{i \in [n]: x_i \neq 0\}$ and $\sigma$ is a positive
integer. The object of investigation is a natural relaxation where we replace
$| \text{supp}(x) | \leq \sigma$ by $\sum_i x_i \leq \sigma$. Our analysis
includes a probabilistic view on when the relaxation is exact. We also consider
the problem from a deterministic point of view and provide a bound on the
distance between the images of optimal solutions of the original problem and
its relaxation under $A$. This leads to an algorithm for generic matrices $A
\in \mathbb{Z}^{m \times n}$ and achieves a polynomial running time provided
that $m$ and $\|A\|_{\infty}$ are fixed.
