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. 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 first gives a fairly complete picture of the computational complexity of this optimization problem, its relation to optimiztion over the core itself, and its equivalence to other, minimal core relaxations that have been proposed earlier. We then address minimum cost spanning tree (MST) games as an example for a class of cost sharing games with non-empty core. While submodular cost functions yield efficient algorithms to maximize shareable costs, MST games have cost functions that are subadditive, but generally not submodular. Nevertheless, it is well known that cost shares in the core of MST games can be found efficiently. In contrast, we show that the maximization of shareable costs is NP-hard for MST games and derive a 2-approximation algorithm. Our work opens several directions for future research.

翻译：本文研究在满足合作可转移效用博弈（简称TU博弈）核心约束的稳定性条件下，最大化一组智能体之间可分摊总成本的优化问题。在最大化可分摊成本时，成本分摊必须满足定义TU博弈核心的所有约束（预算平衡约束除外）。本文首先全面刻画了该优化问题的计算复杂度、其与核心自身优化的关系，以及其与早期提出的其他最小核心松弛问题的等价性。随后，我们以最小成本生成树博弈为例，研究一类具有非空核心的成本分摊博弈。虽然子模成本函数能够通过高效算法最大化可分摊成本，但MST博弈的成本函数具有次可加性，通常不具有子模性。然而，众所周知MST博弈核心中的成本分摊可以高效求得。与之相反，我们证明对于MST博弈，最大化可分摊成本是NP-hard问题，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。