Combinatorial Pen Testing (or Consumer Surplus of Deferred-Acceptance Auctions)

Pen testing is the problem of selecting high-capacity resources when the only way to measure the capacity of a resource expends its capacity. We have a set of $n$ pens with unknown amounts of ink and our goal is to select a feasible subset of pens maximizing the total ink in them. We are allowed to learn about the ink levels by writing with them, but this uses up ink that was previously in the pens. We identify optimal and near optimal pen testing algorithms by drawing analogues to auction theoretic frameworks of deferred-acceptance auctions and virtual values. Our framework allows the conversion of any near optimal deferred-acceptance mechanism into a near optimal pen testing algorithm. Moreover, these algorithms guarantee an additional overhead of at most $(1+o(1)) \ln n$ in the approximation factor to the omniscient algorithm that has access to the ink levels in the pens. We use this framework to give pen testing algorithms for various combinatorial constraints like matroid, knapsack, and general downward-closed constraints, and also for online environments.