facts

Let f be a non-negative submodular function on [n] that is bounded above by 1. First, suppose that Xi are monotone increasing. Construct a sequence X0i as follows. If i / I then set X0i = X0i 1. If i I then set X0i = X0i 1 (Xi \Xi 1). For the monotone decreasing case, consider the submodular function g(X) = f([n] X) and set Yi = [n] Xi.