<p>This paper considers under what conditions an optimal solution to an NP-hard integer linear program can be obtained via a weighted linear programming problem. It is proved that rounding up any optimal solution of a sparse optimization problem results in an optimal solution of the integer linear program. Then, based on the weighted range space property definition, a necessary and sufficient condition is deduced for the uniqueness of an optimal solution of the weighted linear programming problem. Based on the condition, a necessary condition is proposed for the weighted linear programming problem to recover an optimal solution to the sparse optimization problem. Further, we define the weighted range space property of order <i>k</i> and give a sufficient condition for the weighted linear programming problem to exactly recover an optimal solution of the sparse optimization problem, thus obtaining an optimal solution of the integer linear program.</p>