Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Water transport on finite graphs

Timo Vilkas

TL;DR

This work studies how to maximize the water level at a fixed vertex $v$ in a finite graph by sequentially opening pipes, formalizing the objective as $\kappa(v)$. It builds a duality with the Sharing a Drink (SAD) process and introduces meta-sequences and hypermoves to prove that a finite hypermove plan can achieve $\kappa(v)$ (Theorem \text{finitemacro}). The paper establishes NP-hardness for the general problem via a 3-SAT reduction, while also deriving explicit, efficient computations for tractable graphs such as paths and complete graphs. It also develops heuristic approaches (GLA-based) and demonstrates their limitations, outlining open questions about approximation guarantees, optimal move lengths, and behavior on random or fault-prone networks.

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.

Water transport on finite graphs

TL;DR

This work studies how to maximize the water level at a fixed vertex in a finite graph by sequentially opening pipes, formalizing the objective as . It builds a duality with the Sharing a Drink (SAD) process and introduces meta-sequences and hypermoves to prove that a finite hypermove plan can achieve (Theorem \text{finitemacro}). The paper establishes NP-hardness for the general problem via a 3-SAT reduction, while also deriving explicit, efficient computations for tractable graphs such as paths and complete graphs. It also develops heuristic approaches (GLA-based) and demonstrates their limitations, outlining open questions about approximation guarantees, optimal move lengths, and behavior on random or fault-prone networks.

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 and consider the optimization problem relating to the maximum value to which the level in 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.

Paper Structure

This paper contains 13 sections, 8 theorems, 33 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Related work
  4. Preliminaries
  5. Sharing a drink
  6. Meta-sequences and hypermoves
  7. Hypermoves turn supremum into maximum
  8. Algorithmic considerations
  9. Heuristics
  10. Heuristics can be far from optimal
  11. Complexity of the problem
  12. Tractable graphs
  13. Observations and open problems

Key Result

Lemma 2.1

Consider an initial water profile $\{\eta_0(u)\}_{u\in V}$ on $G=(V,E)$ and a finite move sequence $\varphi$ of length $t\in\mathbb{N}$. Then it holds for all $v\in V$ that where $\xi_{u,v}(t)$ is the value at vertex $u$ in the terminal profile of the SAD-process initiated from vertex $v$ with respect to the given move sequence in reversed (time) order, i.e. $\overleftarrow{\varphi}=((e_t,\mu_t),

Figures (13)

  • Figure 1: Converging water levels after opening a lock.
  • Figure 2: An illustration of the pointers from a list, say $L_1=(e_1,e_2,e_1,\dots)$ to the elements of a meta-sequence $\Phi_1$, with for instance $\mathcal{S}(\Phi_1)=\{e_1,e_2,e_7\}$, $\mathcal{N}_1=\{1,2,5,\dots\}$ and $\varphi^{(1)}=(e_2,e_7,e_1,e_1,e_7, e_2, e_7, e_2, e_1, e_2,e_2,\dots)$ etc.
  • Figure 3: The GLA for target vertex $v$ on the left is $\{v,u,\alpha\}$ with value $\mathrm{GLA}(v)=0.6$, and the bottleneck $u$ can be improved by first opening the pipe $\langle u,\beta\rangle$. The GLA for $v$ with respect to the water profile on the right is $\{v\}$, but can be enlarged to $\{v,u,\alpha\}$ if the potential bottleneck $u$ is improved by opening the pipe $\langle u,\beta\rangle$ first.
  • Figure 4: The GLA for target vertex $v$ on the left is $\{v,u,\alpha\}$ with value $\mathrm{GLA}(v)=0.5$. Improving the bottleneck $u$ can be done using either $\beta$ or $\gamma$ and is most effective if the pipe $\langle u,\beta\rangle$ is opened first, then $\langle u,\gamma\rangle$. The GLA for $v$ with respect to the graph on the right is $C=\{v,u,w,\alpha,\beta\}$ with value $\mathrm{GLA}(v)=0.58$. The water from $\gamma$ can be used to improve both bottlenecks $u$ and $w$. It is optimal to open pipe $\langle w,\gamma\rangle$ first and then $\langle u,\gamma\rangle$, raising the average water level in $C$ to 0.62.
  • Figure 5: In a star graph, one can get $\kappa(v)\gg\text{GLA}(v)$.
  • ...and 8 more figures

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Lemma 2.1: Duality
  • Lemma 2.2
  • Lemma 2.3
  • Definition 3
  • Lemma 2.4
  • Example 2.1
  • Definition 4
  • Lemma 2.5
  • ...and 13 more