Inapproximability of truthful mechanisms via generalizations of the Vapnik--Chervonenkis dimension

Algorithmic mechanism design (AMD) studies the delicate interplay between computational efficiency, truthfulness, and optimality. We focus on AMD’s paradigmatic problem: combinatorial auctions. We present a new generalization of the Vapnik–Chervonenkis (VC) dimension to multivalued collections of functions, which encompasses the classical VC dimension, Natarajan dimension, and Steele dimension. We present a corresponding generalization of the Sauer–Shelah lemma and harness this VC machinery to establish inapproximability results for deterministic truthful mechanisms. Our results essentially unify all inapproximability results for deterministic truthful mechanisms for combinatorial auctions to date and establish new separation gaps between truthful and nontruthful algorithms.