Same bang, fewer bucks: efficient discovery of the cost-influence skyline

1 Proofs of propositions Proof. (of Proposition 4.1) Let Xi = X InfMax i for short. First, observe that X1 ∈ P1. This is because the greedy algorithm maximizes marginal gain, and no point on P1 can dominate this, no matter what the costs are. Next, suppose we have Xi−1 ∈ Pi−1. Since Xi is obtained from Xi−1 by adding that u that maximizes marginal gain in I(·), no point on Pi can dominate Xi. This is because for some X ′ ∈ Pi to dominate Xi, there would have to have been some set in Pi−1, such that when some item u′ is added to this set, the marginal gain is larger than the one found by greedy when forming Xi. Because for a submodular I(·) the marginal gains are decreasing, such a u′ can not exist. If it existed, the greedy algorithm would have added u′ at some earlier point.