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

TL;DR

This paper demonstrates a fundamental separation in communication complexity between truthful and non-truthful combinatorial auctions, showing that truthful mechanisms require exponentially more communication for certain approximation guarantees.

Contribution

It proves the first exponential communication lower bound for truthful combinatorial auctions achieving near 3/4 approximation, contrasting with known polynomial bounds for non-truthful algorithms.

Findings

Truthful mechanisms need exponential communication for certain approximation guarantees.

Non-truthful algorithms can achieve 3/4-approximation with polynomial communication.

Lower bounds extend to all truthful mechanisms via taxation complexity framework.

Abstract

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).