We introduce unique sink orientations of grids as digraph models for many well-studied problems, including linear programming over products of simplices, generalized linear complementarity problems over P-matrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of such orientations and develop randomized algorithms for finding the sink. We show that the orientations arising from PGLCP satisfy the Holt-Klee condition known to hold for polytope digraphs, and we

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... e the first expected linear-time algorithms for solving PGLCP with a fixed number of blocks. Keywords Unique sink orientation • Linear programming • Generalized linear complementarity problem • Sink finding algorithm • Holt Klee condition 1 Introduction A grid is a graph whose vertex set is the Cartesian product of n finite sets, with edges joining all pairs of vertices that differ in exactly one component, see Fig. 1a .