In this paper, we present a predictor-corrector interior-point algorithm for $P_{\ast }(\kappa )$-weighted linear complementarity problems. Based on the kernel function $\phi (t)=\sqrt{t}$, the search direction of the algorithm is obtained. By choosing appropriate parameters, we prove that the algorithm is feasible and convergent. It is shown that the proposed algorithm has polynomial iteration complexity. Numerical results illustrate the effectiveness of the algorithm.