Separating the Communication Complexity of Truthful and Non-Truthful Combinatorial Auctions

We provide the first separation in the approximation guarantee achievable by truthful and non-truthful combinatorial auctions with polynomial communication. Specifically, we prove that any truthful mechanism guaranteeing a $(frac{3}{4}-frac{1}{240}+varepsilon)$-approximation for two buyers with XOS valuations over $m$ items requires $exp(Omega(varepsilon^2 cdot m))$ communication, whereas a non-truthful algorithm by Dobzinski and Schapira [SODA 2006] and Feige [2009] is already known to achieve a $frac{3}{4}$-approximation in $poly(m)$ communication. We obtain our separation by proving that any {simultaneous} protocol ({not} necessarily truthful) which guarantees a $(frac{3}{4}-frac{1}{240}+varepsilon)$-approximation requires communication $exp(Omega(varepsilon^2 cdot m))$. The taxation complexity framework of Dobzinski [FOCS 2016] extends this lower bound to all truthful mechanisms (including interactive truthful mechanisms).