This paper presents a new algorithm for the solution of Linear Complementarity Problems (LCP) involving tridiagonal Minkowski matrices. This is an important problem and occurs as a subproblem in two general LCP algorithms discussed by the author. Two algorithms for the tridiagonal Minkowski subproblem are discussed in this paper: one due to Cottle and Sacher [1] and a new algorithm presented in this paper. The new algorithm is presented along with theoretical support, data organization, and implementation considerations. The paper includes a thorough analysis of the computational effort required to solve the problem by both algorithms. The algorithms are numerically compared on two test problems which arise from a free boundary partial journal bearing problem.