Truthful allocation in graphs and hypergraphs

TL;DR

This paper develops new truthful mechanisms for allocation problems in graphs and hypergraphs, improving upon existing methods and providing bounds for various graph classes and objectives.

Contribution

It introduces a novel class of truthful mechanisms with better performance than affine minimizers for multidimensional mechanism design in graph-based allocation problems.

Findings

New upper and lower bounds for truthful mechanisms in multigraphs.

Mechanisms tailored for special graph classes like stars, trees, and planar graphs.

Extension of results to minimizing or maximizing the L^p-norm of players' values.

Abstract

We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of mechanisms that are truthful and have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, $k$-degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the $L^p$-norm of the values of the players, a generalization of the makespan minimization that corresponds to $p=\infty$, and extend the results to any $p>0$.