  In this paper, we propose new efficient algorithms to verify the null space
condition in compressed sensing (CS). Given an $(n-m) \times n$ ($m>0$) CS
matrix $A$ and a positive $k$, we are interested in computing $\displaystyle
\alpha_k = \max_{\{z: Az=0,z\neq 0\}}\max_{\{K: |K|\leq k\}}$ ${\|z_K
\|_{1}}{\|z\|_{1}}$, where $K$ represents subsets of $\{1,2,...,n\}$, and $|K|$
is the cardinality of $K$. In particular, we are interested in finding the
maximum $k$ such that $\alpha_k < {1}{2}$. However, computing $\alpha_k$ is
known to be extremely challenging. In this paper, we first propose a series of
new polynomial-time algorithms to compute upper bounds on $\alpha_k$. Based on
these new polynomial-time algorithms, we further design a new sandwiching
algorithm, to compute the \emph{exact} $\alpha_k$ with greatly reduced
complexity. When needed, this new sandwiching algorithm also achieves a smooth
tradeoff between computational complexity and result accuracy. Empirical
results show the performance improvements of our algorithm over existing known
methods; and our algorithm outputs precise values of $\alpha_k$, with much
lower complexity than exhaustive search.
