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 uni-modularity 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.

DOI: 10.1287/moor.23.1.61

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@article{Cunningham1998IntegralSO, title={Integral Solutions of Linear Complementarity Problems}, author={William H. Cunningham and James F. Geelen}, journal={Math. Oper. Res.}, year={1998}, volume={23}, pages={61-68} }