Water transport on graphs

TL;DR

This paper investigates the problem of redistributing water levels across graph nodes through locked pipes, analyzing its complexity and behavior on various graph types, including finite, infinite, deterministic, and random initial conditions.

Contribution

It introduces the water transport problem on graphs, explores its algorithmic complexity, and compares behaviors on different graph structures, including infinite and finite cases.

Findings

Infinite line graph behaves similarly to finite graphs in water redistribution.

The problem is algorithmically complex and related to opinion formation processes.

Analysis covers both deterministic and random initial water levels.

Abstract

If the nodes of a graph are considered to be identical barrels - featuring different water levels - and the edges to be (locked) water-filled pipes in between the barrels, one might consider the optimization problem of how much the water level in a fixed barrel can be raised with no pumps available, i.e. by opening and closing the locks in an elaborate succession. This 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 algorithmically complex. We deal with both finite and infinite graphs as well as deterministic and random initial water levels and find that the infinite line graph, due to its leanness, behaves much more like a finite graph in this respect.