Beyond Hirsch Conjecture: Walks on Random Polytopes and Smoothed Complexity of the Simplex Method

The smoothed analysis of algorithms is concerned with the expected running time of an algorithm under slight random perturbations of arbitrary inputs. Spielman and Teng proved that the shadow vertex simplex method has polynomial smoothed complexity. On a slight random perturbation of an arbitrary linear program, the simplex method finds the solution after a walk on polytope(s) with expected length polynomial in the number of constraints $n$, the number of variables $d$, and the inverse standard deviation of the perturbation $1/\sigma$. We show that the length of walk in the simplex method is actually polylogarithmic in the number of constraints $n$. Spielman-Teng's bound on the walk was $O^*(n^{86}d^{55}\sigma^{-30})$, up to logarithmic factors. We improve this to $O(\log^7n(d^9+d^3\sigma^{-4}))$. This shows that the tight Hirsch conjecture $n-d$ on the length of walk on polytopes is not a limitation for the smoothed linear programming. Random perturbations create short paths between vertices. We propose a randomized Phase-I for solving arbitrary linear programs, which is of independent interest. Instead of finding a vertex of a feasible set, we add a vertex at random to the feasible set. This does not affect the solution of the linear program with constant probability. So, in expectation it takes a constant number of independent trials until a correct solution is found. This overcomes one of the major difficulties of smoothed analysis of the simplex method—one can now statistically decouple the walk from the smoothed linear program. This yields a much better reduction of the smoothed complexity to a geometric quantity—the size of planar sections of random polytopes. We also improve upon the known estimates for that size, showing that it is polylogarithmic in the number of vertices.