Polynomial-Time Algorithms for Computing the Nucleolus: An Assessment
Holger I. Meinhardt
TL;DR
This work critically evaluates claims of a strongly polynomial-time nucleolus algorithm for convex games by Maggiorano et al., arguing that a misapplication of the Davis/Maschler reduced-game property undermines correctness. It contrasts ellipsoid-method-based approaches with a Fenchel-Moreau conjugation framework that computes a pre-kernel element (and hence the pre-nucleolus when the pre-kernel is single-valued) in polynomial time, specifically $O(n^{3})$ per mapped step. The paper also highlights limitations of reduced-game/SFM strategies that attempt to avoid the ellipsoid method, showing they incur high runtime (roughly $\tilde{O}(n^{7})$–$\tilde{O}(n^{8})$) and practical infeasibility. Overall, the Fenchel-Moreau approach is presented as a principled, scalable alternative for key classes of games, with replication and stability results reinforcing its theoretical and practical appeal.
Abstract
Recently, Maggiorano et al. (2025) claimed that they have developed a strongly polynomial-time combinatorial algorithm for the nucleolus in convex games that is based on the reduced game approach and submodular function minimization method. Thereby, avoiding the ellipsoid method with its negative side effects in numerical computation completely. However, we shall argue that this is a fallacy based on an incorrect application of the Davis/Maschler reduced game property (RGP). Ignoring the fact that despite the pre-nucleolus, other solutions like the core, pre-kernel, and semi-reactive pre-bargaining set possess this property as well. This causes a severe selection issue, leading to the failure to compute the nucleolus of convex games using the reduced games approach. In order to assess this finding in its context, the ellipsoid method of Faigle et al. (2001) and the Fenchel-Moreau conjugation-based approach from convex analysis of Meinhardt (2013) to compute a pre-kernel element were resumed. In the latter case, it was exploited that for TU games with a single-valued pre-kernel, both solution concepts coincide. Implying that one has computed the pre-nucleolus if one has found the sole pre-kernel element of the game. Though it is a specialized and highly optimized algorithm for the pre-kernel, it assures runtime complexity of O(n^3) for computing the pre-nucleolus whenever the pre-kernel is a single point, which indicates a polynomial-time algorithm for this class of games.