The nucleolus is a central solution concept in cooperative game theory. While its computation is NP-hard in general, it can be computed in polynomial time for convex games; however, the only published polynomial-time algorithm relies on the ellipsoid method. We develop a combinatorial alternative based on reduced games and iterative least-core value computations. Leveraging submodular function minimization and polyhedral structure in a novel way, we obtain a faster combinatorial algorithm for computing the least-core value, improving the oracle complexity by a factor $n^3$ over previous approaches. As a consequence, we obtain a new strongly polynomial-time and combinatorial algorithm for computing the nucleolus in convex games. Preliminary analysis indicates an improved oracle complexity compared to the ellipsoid-based algorithm.

翻译：核仁是合作博弈论中的核心解概念。尽管其计算在一般情况下是NP难的，但在凸博弈中可在多项式时间内计算；然而，目前唯一已发表的多项式时间算法依赖于椭球法。我们基于约简博弈和迭代最小核心值计算，提出了一种组合替代算法。通过创新性地利用次模函数最小化和多面体结构，我们得到了一种计算最小核心值的更快组合算法，将预言机复杂度较先前方法提升了$n^3$倍。由此，我们获得了一种新的强多项式时间组合算法，用于计算凸博弈中的核仁。初步分析表明，该算法相比基于椭球法的算法具有更优的预言机复杂度。