# Algorithmic Solutions for Maximizing Shareable Costs

## TL;DR

This paper investigates the computational complexity of maximizing shareable costs in cooperative games, showing NP-hardness for certain cases and providing approximation algorithms, with implications for cost sharing in network design.

## Contribution

It characterizes the complexity of maximizing shareable costs in TU games and introduces a 2-approximation algorithm for MST games where this problem is NP-hard.

## Key findings

- Maximizing shareable costs is NP-hard for MST games.
- A 2-approximation algorithm is proposed for this problem.
- Submodular functions allow efficient maximization, unlike MST games.

## Abstract

This paper addresses the optimization problem to maximize the total costs that can be shared among a group of agents, while maintaining stability in the sense of the core constraints of a cooperative transferable utility game, or TU game. When maximizing total shareable costs, the cost shares must satisfy all constraints that define the core of a TU game, except for being budget balanced. The paper first gives a fairly complete picture of the computational complexity of this optimization problem, its relation to optimiztion over the core itself, and its equivalence to other, minimal core relaxations that have been proposed earlier. We then address minimum cost spanning tree (MST) games as an example for a class of cost sharing games with non-empty core. While submodular cost functions yield efficient algorithms to maximize shareable costs, MST games have cost functions that are subadditive, but generally not submodular. Nevertheless, it is well known that cost shares in the core of MST games can be found efficiently. In contrast, we show that the maximization of shareable costs is NP-hard for MST games and derive a 2-approximation algorithm. Our work opens several directions for future research.