In this paper we extend primal-dual interior point algorithm for linear optimization (LO) problems to $P_*({kappa})$ linear complementarity problems(LCPs) ([1]). We define proximity functions and search directions based on kernel functions, ${psi}(t)=frac{t^{p+1}-1}{p+1}-{log};t$, $p{in}$[0, 1], which is a generalized form of the one in [16]. It is the first to use this class of kernel functions in the complexity analysis of interior point method(IPM) for $P_*({kappa})$ LCPs. We show that if a strictly feasible starting point is available, then new large-update primal-dual interior point algorithms for $P_*({kappa})$ LCPs have $O((1+2{kappa})nlog{frac{n}{varepsilon}})$ complexity which is similar to the one in [16]. For small-update methods, we have $O((1+2{kappa})sqrt{n}{log}{frac{n}{varepsilon}})$ which is the best known complexity so far.