Arboricity games: the core and the nucleolus

TL;DR

This paper introduces a game-theoretic approach to graph arboricity, defining the arboricity game, analyzing the core and nucleolus, and providing an efficient polynomial-time algorithm for computing the nucleolus using a novel prime partition.

Contribution

It proposes the arboricity game model, introduces the prime partition for graph decomposition, and offers a polynomial-time algorithm for nucleolus computation when the core is non-empty.

Findings

The prime partition reduces complexity in nucleolus computation.

The core and nucleolus properties are characterized for the arboricity game.

A polynomial-time algorithm for nucleolus computation is developed.

Abstract

The arboricity of a graph is the minimum number of forests required to cover all its edges. In this paper, we examine arboricity from a game-theoretic perspective and investigate cost-sharing in the minimum forest cover problem. We introduce the arboricity game as a cooperative cost game defined on a graph. The players are edges, and the cost of each coalition is the arboricity of the subgraph induced by the coalition. We study properties of the core and propose an efficient algorithm for computing the nucleolus when the core is not empty. In order to compute the nucleolus in the core, we introduce the prime partition which is built on the densest subgraph lattice. The prime partition decomposes the edge set of a graph into a partially ordered set defined from minimal densest minors and their invariant precedence relation. Moreover, edges from the same partition always have the same value in a core allocation. Consequently, when the core is not empty, the prime partition significantly reduces the number of variables and constraints required in the linear programs of Maschler's scheme and allows us to compute the nucleolus in polynomial time. Besides, the prime partition provides a graph decomposition analogous to the celebrated core decomposition and the density-friendly decomposition, which may be of independent interest.