In this talk, we consider computing a nonzero solution to the linear complementarity problem $0 \leq x \;\perp\; Mx + q \geq 0$ where $q > 0$. We present several formulations of this problem and their properties including one reformulation with a strictly positive matrix for which Lemke's method with appropriate degeneracy resolution is guaranteed to compute a solution. We then apply the standard Lemke method with a ray start as implemented in PATH to random sparse instances from this problem class and detect physical cycles of nonzero length in the path constructed. We then discuss randomization methods to address the degeneracy and show that these problem can be effectively solved.