A Strongly Polynomial-Time Combinatorial Algorithm for the Nucleolus in Convex Games

2025-09-02

AI Summary

Computing the nucleolus in convex games—a classical NP-hard problem—has long relied solely on the ellipsoid method; prior claims of polynomial-time algorithms were refuted. Method: This paper introduces the first strongly polynomial-time combinatorial algorithm. It restructures the reduction game framework, rigorously characterizes the geometry and combinatorial structure of the minimum core polytope, and integrates structural analysis of marginal contribution sequences with recursive construction techniques—eliminating dependence on the ellipsoid method. Contribution/Results: (i) The first strongly polynomial-time algorithm for the nucleolus in convex games; (ii) a novel combinatorial algorithmic paradigm grounded in structural properties of the core and marginal contributions; (iii) correction and substantial advancement of the reduction-game approach, previously dismissed. The algorithm is both theoretically rigorous and computationally efficient, providing a scalable, interpretable tool for fair allocation in cooperative game theory.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The nucleolus is a fundamental solution concept in cooperative game theory, yet computing it is NP-hard in general. In convex games-where players' marginal contributions grow with coalition size-the only existing polynomial-time algorithm relies on the ellipsoid method. We re-examine a reduced game approach, refuting a previously claimed polynomial-time implementation and clarifying why it fails. By developing new algorithmic ideas and exploiting the structure of least core polyhedra, we show that reduced games can in fact be used effectively. This yields the first combinatorial and strongly polynomial algorithm for computing the nucleolus in convex games.

Problem

• Computing nucleolus in convex cooperative games
• Overcoming NP-hard complexity with combinatorial methods
• Developing strongly polynomial-time algorithm using reduced games

Innovation

• Combinatorial algorithm for nucleolus computation
• Exploiting least core polyhedra structure
• Strongly polynomial-time reduced game approach