This paper addresses the optimization problem to maximize the total costs that can be shared among a group of agents, while maintaining stability in the sense of the core constraints of a cooperative transferable utility game, or TU game. This means that all subsets of agents have an outside option at a certain cost, and stability requires that the cost shares are defined so that none of the outside options is preferable. When maximizing total shareable costs, the cost shares must satisfy all constraints that define the core of a TU game, except for being budget balanced. The paper gives a fairly complete picture of the computational complexity of this optimization problem, in relation to classical computational problems on the core. We also show that, for games with an empty core, the problem is equivalent to computing minimal core relaxations for several relaxations that have been proposed earlier. As an example for a class of cost sharing games with non-empty core, we address minimum cost spanning tree games. While it is known that cost shares in the core can be found efficiently, we show that the computation of maximal cost shares is NP-hard for minimum cost spanning tree games. We also derive a 2-approximation algorithm. Our work opens several directions for future work.

翻译：摘要：本文研究了在合作可转移效用博弈（TU博弈）的核心约束下，最大化一组代理之间可分摊总成本的优化问题，同时保持稳定性。这意味着所有代理子集在特定成本下拥有外部选择，稳定性要求成本分摊的定义确保任何外部选择都不具优势。在最大化总可分摊成本时，成本分摊需满足定义TU博弈核心的所有约束，唯预算平衡约束除外。本文对此优化问题的计算复杂度给出了较为完整的图景，并与核心上的经典计算问题相关联。我们还证明，对于核心为空的博弈，该问题等价于计算针对先前提出的若干松弛方法的最小核心松弛。以一类非空核心的成本分摊博弈为例，我们研究了最小成本生成树博弈。虽然已知在核心中高效计算成本分摊的方法，但我们证明，对于最小成本生成树博弈，最大成本分摊的计算是NP难的，并推导出一个2-近似算法。我们的工作为未来研究开辟了多个方向。