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 $p>0$.

中文翻译：

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图和超图中的真实分配

我们研究了图中分配问题的真实机制，包括最小化（即调度）和最大化（即拍卖）设置。最小化问题是经过充分研究的无关机器调度问题的一个特例，其中每个给定的任务在图的情况下只能由两个预先指定的机器或在超图的情况下的给定机器子集执行。这对应于一个多重图，其节点是机器，其超边是任务。这类问题属于多维机制设计，除 VCG 及其对仿射最小化器的泛化外，没有其他已知的通用机制。我们提出了一类新的机制，这些机制是真实的，并且在许多设置中具有比仿射最小化器更好的性能。具体来说，我们为一般多重图以及特殊类别的图（例如星形、树、平面图、$k$-退化图和给定树宽的图）的真实机制提供了上限和下限。我们还考虑了最小化或最大化玩家价值的 $L^p$-范数的目标，对应于 $p=\infty$ 的 makespan 最小化的概括，并将结果扩展到任何 $p>0 $.