Strategic deployment in graphs

TL;DR

This paper introduces a new graph-based deployment problem inspired by historical military strategies, providing polynomial solutions for trees and approximation methods for general graphs.

Contribution

It formulates the off-line graph deployment problem, proves NP-hardness for general graphs, and offers efficient algorithms for trees and approximation strategies for arbitrary graphs.

Findings

Optimal deployment for trees can be computed in polynomial time.

Minimum spanning tree solutions give a 2-approximation for general graphs.

The problem is NP-hard for general graphs.

Abstract

Conquerors of old (like, e.g., Alexander the Great or Ceasar) had to solve the following deployment problem. Sufficiently strong units had to be stationed at locations of strategic importance, and the moving forces had to be strong enough to advance to the next location. To the best of our knowledge we are the first to consider the (off-line) graph version of this problem. While being NP-hard for general graphs, for trees the minimum number of agents and an optimal deployment can be computed in optimal polynomial time. Moreover, the optimal solution for the minimum spanning tree of an arbitrary graph G results in a 2-approximation of the optimal solution for G.