Polynomial-Time Algorithms for Computing the Nucleolus: An Assessment

TL;DR

This paper critically assesses recent claims of a polynomial-time algorithm for the nucleolus in convex games, highlighting fundamental errors and discussing alternative approaches with proven polynomial complexity.

Contribution

It refutes the claim of a polynomial-time algorithm for the nucleolus based on the reduced game approach and discusses alternative methods with established polynomial runtime.

Findings

The claimed algorithm is based on an incorrect application of the reduced game property.

Alternative methods like the ellipsoid and convex analysis approaches are discussed.

A polynomial-time algorithm exists for the pre-nucleolus in certain game classes.

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.