  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 $p>0$.
