Water transport on finite graphs

TL;DR

This paper investigates an optimization problem on finite graphs involving water levels in connected barrels, which is complex and has applications in opinion formation models.

Contribution

It introduces a novel optimization problem on graphs related to water transport and analyzes its computational complexity.

Findings

The problem is computationally challenging (likely NP-hard).

It has applications in modeling opinion dynamics.

The problem's structure offers insights into related network flow issues.

Abstract

Consider a simple finite graph and its nodes to represent identical water barrels (containing different amounts of water) on a level plane. Each edge corresponds to a (locked, water-filled) pipe connecting two barrels below the plane. We fix one node $v$ and consider the optimization problem relating to the maximum value to which the level in $v$ can be raised without pumps, i.e. by opening/closing pipes in a suitable order. This fairly natural optimization problem originated from the analysis of an opinion formation process and proved to be not only sufficiently intricate in order to be of independent interest, but also difficult from an algorithmic point of view.