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In this paper, we study the sparsest solutions of linear complementarity problems (LCPs), which study has many applications, such as bimatrix games and portfolio selections. Mathematically, the underlying model is NP-hard in general. By transforming the complementarity constraints into a fixed point equation with projection type, we propose an l1 regularization projection minimization model for relaxation. Through developing a thresholding representation of solutions for a key subproblem of this regularization model, we design a shrinkage-thresholding projection (STP) algorithm to solve this model and also analyze convergence of STP algorithm. Numerical results demonstrate that the STP method can efficiently solve this regularized model and get a sparsest solution of LCP with high quality. [ABSTRACT FROM AUTHOR]