The Linear Complementarity Problem, Lemke Algorithm, Perturbation, and the Complexity Class PPAD

We present a single sufficient condition for the processability of the Lemke algorithm for semimonotone Linear Complementarity problems (LCP) which unifies several sufficient conditions for a number of well known subclasses of semimonotone LCPs. In particular, we study the close relationship of these problems to the complexity class PPAD. Next, we show that these classes of LCPs can be reduced (polynomially), by perturbation, to stricter sub-classes and establish an alternative (non-combinatorial) scheme to prove their membership in PPAD. We then identify several of these subclasses as PPAD-complete and discuss the likelihood that the other subclasses (all of which are reducible to LCP’s with P matrices) are PPAD-complete. Finally, we present an hierarchy of subclasses of the semimonotone LCP’s within the complexity classes of P, PPAD and NP-complete.