Improved upper bounds for random-edge and random-jump on abstract cubes

Upper bounds are given for the complexity of two very natural randomized algorithms for finding the sink of an Acyclic Unique Sink Orientation (AUSO) of the n-cube. For Random-Edge, we obtain an upper bound of about 1.80n, improving upon the the previous upper bound of about 2n/nlog n obtained by Gärtner and Kaibel. For Random-Jump, we obtain an upper bound of about (3/2)n, improving upon the previous upper bound of about 1.72n obtained by Mansour and Singh. AUSOs provide an appealing combinatorial abstraction of linear programming and other computational problems such as finding optimal strategies for turn-based Stochastic Games.