An $O(\log \log n)$-approximate budget feasible mechanism for subadditive valuations

TL;DR

This paper introduces a randomized polynomial-time mechanism for budget-feasible procurement with subadditive valuations, achieving an improved approximation factor of O(log log n) over previous methods.

Contribution

The paper presents the first mechanism with an O(log log n) approximation for subadditive valuations, surpassing the prior O(log n / log log n) bound.

Findings

Achieves O(log log n) approximation factor.

Runs in polynomial time with demand oracle access.

Significantly improves previous approximation bounds.

Abstract

In budget-feasible mechanism design, there is a set of items $U$, each owned by a distinct seller. The seller of item $e$ incurs a private cost $\overline{c}_e$ for supplying her item. A buyer wishes to procure a set of items from the sellers of maximum value, where the value of a set $S\subseteq U$ of items is given by a valuation function $v:2^U\to \mathbb{R}_+$. The buyer has a budget of $B \in \mathbb{R}_+$ for the total payments made to the sellers. We wish to design a mechanism that is truthful, that is, sellers are incentivized to report their true costs, budget-feasible, that is, the sum of the payments made to the sellers is at most the budget $B$, and that outputs a set whose value is large compared to $\text{OPT}:=\max\{v(S):\overline{c}(S)\le B,S\subseteq U\}$. Budget-feasible mechanism design has been extensively studied, with the literature focussing on (classes of) subadditive valuation functions, and various polytime, budget-feasible mechanisms, achieving constant-factor approximation, have been devised for the special cases of additive, submodular, and XOS valuations. However, for general subadditive valuations, the best-known approximation factor achievable by a polytime budget-feasible mechanism (given access to demand oracles) was only $O(\log n / \log \log n)$, where $n$ is the number of items. We improve this state-of-the-art significantly by designing a randomized budget-feasible mechanism for subadditive valuations that achieves a substantially-improved approximation factor of $O(\log\log n)$ and runs in polynomial time, given access to demand oracles.