This paper shows that the least absolute shrinkage and selection operator (LASSO) can provide an exact optimal solution to a special type of constrained cardinality minimization problem, which is motivated from a sensor network measurement robustness analysis problem. The constraint matrix of the considered problem is totally unimodular. This is shown to imply that LASSO leads to a tight linear programming relaxation of the considered cardinality minimization problem. The presented result is based on a deterministic argument, which differs from other well-known deterministic results for LASSO such as mutual incoherence and restricted isometry property. There exist cases given in this paper where the proposed result is applicable, while mutual incoherence and restricted isometry property are not.