Fast Convex Decomposition for Truthful Social Welfare Approximation

TL;DR

This paper introduces a new convex decomposition method that improves the practicality of designing truthful mechanisms with near-optimal social welfare approximation, reducing reliance on the ellipsoid method.

Contribution

The authors develop an alternative convex decomposition technique that achieves an $oldsymbol{ extit{ extbf{alpha}}}(1 + oldsymbol{ extit{ extbf{ extepsilon}}})$ approximation with fewer calls to the integrality gap verifier, enhancing efficiency.

Findings

Achieves $oldsymbol{ extit{ extbf{alpha}}}(1 + oldsymbol{ extit{ extbf{ extepsilon}}})$ approximation

Requires only quadratic calls to the integrality gap verifier

Improves practical applicability over previous methods

Abstract

Approximating the optimal social welfare while preserving truthfulness is a well studied problem in algorithmic mechanism design. Assuming that the social welfare of a given mechanism design problem can be optimized by an integer program whose integrality gap is at most $\alpha$, Lavi and Swamy~\cite{Lavi11} propose a general approach to designing a randomized $\alpha$-approximation mechanism which is truthful in expectation. Their method is based on decomposing an optimal solution for the relaxed linear program into a convex combination of integer solutions. Unfortunately, Lavi and Swamy's decomposition technique relies heavily on the ellipsoid method, which is notorious for its poor practical performance. To overcome this problem, we present an alternative decomposition technique which yields an $\alpha(1 + \epsilon)$ approximation and only requires a quadratic number of calls to an integrality gap verifier.