Computing the Least-core and Nucleolus for Threshold Cardinality Matching Games

TL;DR

This paper investigates the computational aspects of the least-core and nucleolus in threshold cardinality matching games, providing polynomial algorithms and characterizations for these solution concepts in specific graph classes.

Contribution

It introduces polynomial-time algorithms for computing the least-core and nucleolus in TCMGs and offers new characterizations, especially for the case when the threshold T equals 1.

Findings

Least-core value can be computed in polynomial time.

Nucleolus can be efficiently found in bipartite graphs and graphs with perfect matchings.

Characterizations of the least core for a broad class of TCMGs are provided.

Abstract

Cooperative games provide a framework for fair and stable profit allocation in multi-agent systems. \emph{Core}, \emph{least-core} and \emph{nucleolus} are such solution concepts that characterize stability of cooperation. In this paper, we study the algorithmic issues on the least-core and nucleolus of threshold cardinality matching games (TCMG). A TCMG is defined on a graph $G=(V,E)$ and a threshold $T$, in which the player set is $V$ and the profit of a coalition $S\subseteq V$ is 1 if the size of a maximum matching in $G[S]$ meets or exceeds $T$, and 0 otherwise. We first show that for a TCMG, the problems of computing least-core value, finding and verifying least-core payoff are all polynomial time solvable. We also provide a general characterization of the least core for a large class of TCMG. Next, based on Gallai-Edmonds Decomposition in matching theory, we give a concise formulation of the nucleolus for a typical case of TCMG which the threshold $T$ equals $1$. When the threshold $T$ is relevant to the input size, we prove that the nucleolus can be obtained in polynomial time in bipartite graphs and graphs with a perfect matching.