The Computational Complexity of Truthfulness in Combinatorial Auctions

TL;DR

This paper establishes computational hardness results showing that truthful mechanisms for combinatorial auctions with succinctly described valuations cannot achieve good approximation ratios unless major complexity class collapses occur.

Contribution

It is the first to prove hardness results for polynomial-time truthful mechanisms in combinatorial auctions with succinct valuations, highlighting fundamental limitations.

Findings

No deterministic truthful mechanism approximates within m^{1/2-ε} unless NP=RP.

Truthful-in-expectation mechanisms cannot approximate better than n^γ unless NP⊆P/poly.

Hardness results extend to related auction problems.

Abstract

One of the fundamental questions of Algorithmic Mechanism Design is whether there exists an inherent clash between truthfulness and computational tractability: in particular, whether polynomial-time truthful mechanisms for combinatorial auctions are provably weaker in terms of approximation ratio than non-truthful ones. This question was very recently answered for universally truthful mechanisms for combinatorial auctions \cite{D11}, and even for truthful-in-expectation mechanisms \cite{DughmiV11}. However, both of these results are based on information-theoretic arguments for valuations given by a value oracle, and leave open the possibility of polynomial-time truthful mechanisms for succinctly described classes of valuations. This paper is the first to prove {\em computational hardness} results for truthful mechanisms for combinatorial auctions with succinctly described valuations. We prove that there is a class of succinctly represented submodular valuations for which no deterministic truthful mechanism provides an $m^{1/2-\epsilon}$-approximation for a constant $\epsilon>0$, unless $NP=RP$ ($m$ denotes the number of items). Furthermore, we prove that even truthful-in-expectation mechanisms cannot approximate combinatorial auctions with certain succinctly described submodular valuations better than within $n^\gamma$, where $n$ is the number of bidders and $\gamma>0$ some absolute constant, unless $NP \subseteq P/poly$. In addition, we prove computational hardness results for two related problems.