Greedy and randomized heuristics for optimization of k-domination models in digraphs and road networks

TL;DR

This paper introduces new greedy and randomized heuristics for optimizing k-dominating sets in directed graphs, particularly applied to road networks for refuelling station placement, demonstrating efficiency and effectiveness in large-scale instances.

Contribution

It presents the first reachability digraph model for road networks and develops novel greedy and probabilistic heuristics for k-dominating set optimization.

Findings

Refined greedy heuristics efficiently find small k-dominating sets.

A probabilistic method establishes a new upper bound on the k-domination number.

Heuristics perform well on large real-world road network data.

Abstract

Directed graphs provide more subtle and precise modelling tools for optimization in road networks than simple graphs. In particular, they are more suitable in the context of alternative fuel vehicles and new automotive technologies, like electric vehicles. In this paper, we introduce the new general concept of a reachability digraph associated with a road network to model the placement of refuelling facilities in road networks as k-dominating sets in the reachability digraph. Two new greedy heuristics are designed and experimentally tested to search for small k-dominating sets in two types of digraphs, including the reachability digraphs. Refined greedy strategies are shown to be efficient, capable of finding good quality solutions, and suitable for application in very large digraphs and road networks. Also, a probabilistic method is used to prove a new upper bound on the k-domination number of a digraph, which informs the development of a new randomized heuristic to search for k-dominating sets in the digraph. Generalizing the randomized heuristic ideas, making the heuristic more flexible, tuning and combining it with the greedy strategies allows us to obtain even better results for the reachability digraphs. Computational experiments are conducted for a case study of road networks in the West Midlands (UK).