RANDOM EDGE can be exponential on abstract cubes

Jiří Matoušek, Tibor Szabó
2006 Advances in Mathematics  
We prove that RANDOM EDGE, the simplex algorithm that always chooses a random improving edge to proceed on, can take a mildly exponential number of steps in the model of abstract objective functions (introduced by Williamson Hoke [Completely unimodal numberings of a simple polytope, Discrete Appl. Math. 20 (1988) 69-81.] and by Kalai [A simple way to tell a simple polytope from its graph, J. Combin. Theory Ser. A 49(2) (1988) 381-383.] under different names). We define an abstract objective
more » ... tion on the n-dimensional cube for which the algorithm, started at a random vertex, needs at least exp(const · n 1/3 ) steps with high probability. The best previous lower bound was quadratic. So in order for RANDOM EDGE to succeed in polynomial time, geometry must help.
doi:10.1016/j.aim.2005.05.021 fatcat:mmftd7pxdrbmhc52h7h73ondeu