  The 0-1 linear programming problem with nonnegative constraint matrix and
objective vector e origins from many NP-hard combinatorial optimization
problems. In this paper, we consider recovering an optimal solution to the
problem from a weighted linear programming.We first formulate the problem
equivalently as a sparse optimization problem. Next, we consider the
consistency of the optimal solution of the sparse optimization problem and the
weighted linear programming problem. In order to achieve this, we establish
nonnegative partial s-goodness of the constraint matrix and the weighted
vector. Further, we use two quantities to characterize a sufficient condition
and necessary condition for the nonnegative partial s-goodness. However, the
two quantities are difficult to calculate, therefore, we provide a computable
upper bound for one of the two quantities to verify the nonnegative partial
s-goodness. Finally, we give three examples to illustrate that our theory is
effective and verifiable.
