  We present a new interior-point potential-reduction algorithm for solving
monotone linear complementarity problems (LCPs) that have a particular special
structure: their matrix $M\in{\mathbb R}^{n\times n}$ can be decomposed as
$M=\Phi U + \Pi_0$, where the rank of $\Phi$ is $k<n$, and $\Pi_0$ denotes
Euclidean projection onto the nullspace of $\Phi^\top$. We call such LCPs
projective. Our algorithm solves a monotone projective LCP to relative accuracy
$\epsilon$ in $O(\sqrt n \ln(1/\epsilon))$ iterations, with each iteration
requiring $O(nk^2)$ flops. This complexity compares favorably with
interior-point algorithms for general monotone LCPs: these algorithms also
require $O(\sqrt n \ln(1/\epsilon))$ iterations, but each iteration needs to
solve an $n\times n$ system of linear equations, a much higher cost than our
algorithm when $k\ll n$. Our algorithm works even though the solution to a
projective LCP is not restricted to lie in any low-rank subspace.
