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

Lukas Dijkstra, Andrei Gagarin, Padraig Corcoran, Rhyd Lewis

TL;DR

The paper tackles placing refuelling facilities in road networks by casting the problem as finding small $k$-dominating sets in reachability digraphs, leveraging the directionality of travel and energy considerations. It introduces three greedy heuristics (BG, DCG, TCG) and a probabilistic, randomized method that uses an upper bound on the $k$-domination number to guide search; the RandAlg variant further tunes randomness with degree-based parameters. Across Erdős–Rényi and real road-network reachability digraphs, DCG and especially TCG frequently outperform Basic Greedy, while randomized strategies provide additional gains in many large instances at the cost of higher runtimes. The results demonstrate scalable, implementable techniques for large networks and offer a framework for future enhancements in digraph domination models and their applicability to facility-location problems beyond refuelling, including emergency services. All mathematical notation is presented with proper delimiters to support precise interpretation and reuse in search and optimization workflows.

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).

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

TL;DR

The paper tackles placing refuelling facilities in road networks by casting the problem as finding small -dominating sets in reachability digraphs, leveraging the directionality of travel and energy considerations. It introduces three greedy heuristics (BG, DCG, TCG) and a probabilistic, randomized method that uses an upper bound on the -domination number to guide search; the RandAlg variant further tunes randomness with degree-based parameters. Across Erdős–Rényi and real road-network reachability digraphs, DCG and especially TCG frequently outperform Basic Greedy, while randomized strategies provide additional gains in many large instances at the cost of higher runtimes. The results demonstrate scalable, implementable techniques for large networks and offer a framework for future enhancements in digraph domination models and their applicability to facility-location problems beyond refuelling, including emergency services. All mathematical notation is presented with proper delimiters to support precise interpretation and reuse in search and optimization workflows.

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).
Paper Structure (15 sections, 2 theorems, 14 equations, 6 figures, 7 tables, 5 algorithms)

This paper contains 15 sections, 2 theorems, 14 equations, 6 figures, 7 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Digraphs
  3. Reachability digraphs for vehicles in road networks
  4. Multiple dominating sets in digraphs
  5. Related Work
  6. Heuristic Algorithms and the Upper Bound
  7. Greedy heuristics
  8. Randomized heuristic
  9. Using the probabilistic method
  10. Combining and tuning the randomized and greedy heuristics
  11. Computational Experiments
  12. ILP formulation and experiments with a deterministic ILP solver
  13. Erdős-Rényi digraphs
  14. Road networks of Birmingham and the West Midlands conurbation
  15. Conclusions

Key Result

Theorem 1

Given a simple graph $G$ of minimum vertex degree $\delta=\delta(G)$ and an integer $k$, $1\le k\le \delta$, we have where $\delta^\prime = \delta -k +1$.

Figures (6)

  • Figure 1: A digraph model of a road network showing a vertex (yellow) and its out-neighbours (red) in the corresponding reachability digraph.
  • Figure 2: Basic Greedy
  • Figure 3: Minimal $k$-Dominating Subset
  • Figure 4: Deficiency Coverage Greedy
  • Figure 5: Two-Criteria Greedy
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1: GPZ2013Zve2
  • Theorem 2