Random walks on truncated cubes and sampling 0-1 knapsack solutions

We solve an open problem concerning the mixing time of a symmetric random walk on an n-dimensional cube truncated by a hyperplane, showing that it is polynomial in n. As a consequence, we obtain a full-polynomial randomized approximation scheme for counting the feasible solutions of a 0-1 knapsack problem. The key ingredient in our analysis is a combinatorial construction we call a "balanced almost uniform permutation", which seems to be of independent interest.