Complexity of inheritance of $\mathcal{F}$-convexity for restricted games induced by minimum partitions

TL;DR

This paper investigates the inheritance of $ ext{F}$-convexity in restricted cooperative games derived from weighted graphs, providing a polynomial-time method to determine when this property is preserved.

Contribution

It introduces a polynomial-time decision procedure for inheritance of $ ext{F}$-convexity in $P_{ ext{min}}$-restricted games based on minimum partitions.

Findings

Polynomial-time algorithm for inheritance decision

Characterization of $ ext{F}$-convexity inheritance

Application to restricted cooperative game analysis

Abstract

Let $G = (N,E,w)$ be a weighted communication graph (with weight function $w$ on $E$). For every subset $A \subseteq N$, we delete in the subset $E(A)$ of edges with ends in $A$, all edges of minimum weight in $E(A)$. Then the connected components of the corresponding induced subgraph constitute a partition of $A$ that we call $P_{\min}(A)$. For every game $(N, v)$, we define the $P_{\min}$-restricted game $(N, \bar{v})$ by $\bar{v}(A) = \sum_{F \in P_{\min}(A)} v(F)$ for all $A \subseteq N$. We prove that we can decide in polynomial time if there is inheritance of $\mathcal{F}$-convexity from $(N, v)$ to the $P_{\min}$-restricted game $(N, \bar{v})$ where $\mathcal{F}$-convexity is obtained by restricting convexity to connected subsets.