On the Nucleolus of a Class of Linear Production Games

TL;DR

This paper investigates the nucleolus in a specific subclass of linear production games related to logistics, providing polynomial-time algorithms for certain cases despite the general NP-hardness.

Contribution

It introduces structural properties of production-distribution games that enable efficient nucleolus computation in uncapacitated cases, including fixed markets and single-market scenarios.

Findings

Polynomial-time characterization for core singleton instances

Separation-based polynomial algorithm for fixed markets

Fast combinatorial primal-dual algorithm for single-market case

Abstract

We study the nucleolus in a class of cooperative games where agents collaborate by sharing demands and production-distribution capacities across multiple markets. These production-distribution games form a structured subclass of linear production games and capture applications such as horizontal collaboration in logistics. While computing the nucleolus is generally NP-hard for linear production games, we show that structural properties of production-distribution games enable efficient computation in several cases. Our main results focus on the uncapacitated variant. First, we provide a polynomial-time characterization of instances where the core reduces to a singleton, allowing direct computation of the nucleolus. Second, when the number of markets is fixed, we design a separation-based polynomial-time algorithm. Third, in the single-market case, we develop a faster combinatorial primal-dual algorithm that runs in $O(n^4)$.