Breaking the Logarithmic Barrier for Truthful Combinatorial Auctions with Submodular Bidders

TL;DR

This paper introduces a novel truthful mechanism for combinatorial auctions with submodular bidders that surpasses the previous logarithmic approximation barrier, achieving an $O( oot{ ext{log} m})$ ratio, and extends to budget additive valuations.

Contribution

It presents the first mechanism breaking the logarithmic barrier for approximation ratios in combinatorial auctions with submodular bidders, including an efficient implementation for budget additive valuations.

Findings

Achieves an $O( oot{ ext{log} m})$ approximation ratio.

Works with polynomially many value and demand queries.

Provides an efficient implementation for budget additive bidders.

Abstract

We study a central problem in Algorithmic Mechanism Design: constructing truthful mechanisms for welfare maximization in combinatorial auctions with submodular bidders. Dobzinski, Nisan, and Schapira provided the first mechanism that guarantees a non-trivial approximation ratio of $O(\log^2 m)$ [STOC'06], where $m$ is the number of items. This was subsequently improved to $O(\log m\log \log m)$ [Dobzinski, APPROX'07] and then to $O(\log m)$ [Krysta and Vocking, ICALP'12]. In this paper we develop the first mechanism that breaks the logarithmic barrier. Specifically, the mechanism provides an approximation ratio of $O(\sqrt {\log m})$. Similarly to previous constructions, our mechanism uses polynomially many value and demand queries, and in fact provides the same approximation ratio for the larger class of XOS (a.k.a. fractionally subadditive) valuations. We also develop a computationally efficient implementation of the mechanism for combinatorial auctions with budget additive bidders. Although in general computing a demand query is NP-hard for budget additive valuations, we observe that the specific form of demand queries that our mechanism uses can be efficiently computed when bidders are budget additive.