Integral Solutions of Linear Complementarity Problems

We characterize the class of integral square matrices M having the property that for every integral vector q the linear complementarity problem with data M, q has only integral basic solutions. These matrices, called principally unimodular matrices, are those for which every principal nonsingular submatrix is unimodular. As a consequence, we show that if M is rank-symmetric and principally unimodular, and q is integral, then the problem has an integral solution if it has a solution. Principal unimodularity can be regarded as an extension of total unimodularity, and our results can be regarded as extensions of well-known results on integral solutions to linear programs. We summarize what is known about principally unimodular symmetric and skew-symmetric matrices.