We consider linearly weighted versions of the least core and the (pre)nuceolus and investigate the reduction possibilities in their computation. We slightly extend some well-known related results and establish their counterparts by using the dual game. Our main results imply, for example, that if the core of the game is not empty, all dually inessential coalitions (which can be weakly minorized by a partition in the dual game) can be ignored when we compute the per-capita least core and the per-capita (pre)nucleolus from the dual game. This could lead to the design of polynomial time algorithms for the per-capita (and other monotone nondecreasingly weighted versions of the) least core and the (pre)nucleolus in specific classes of balanced games with polynomial many dually essential coalitions.

中文翻译：

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加权核仁和双重必要联盟

我们考虑最小核心和（前）核仁的线性加权版本，并研究它们计算中的减少可能性。我们稍微扩展了一些众所周知的相关结果，并通过使用对偶博弈建立它们的对应结果。例如，我们的主要结果意味着，如果博弈的核心不为空，那么当我们计算人均最小核心时，所有对偶非本质联盟（可以被对偶游戏中的分区弱化）都可以忽略不计。来自双重博弈的人均（前）核仁。这可能会导致设计多项式时间算法，用于在具有多项式许多双重基本联盟的特定类别的平衡游戏中，为人均（以及其他单调非递减加权版本）最小核心和（前）核仁。