On the -Strong Roman Domination Problem
Graphical abstract
Introduction
The Roman Domination Problem (RDP) has been studied intensively since it was proposed by Revelle and Rosing [25]. According to Luttwak [22], in the fourth century CE, the Roman Empire has adopted a “defense-in-depth” strategy, in order to defend the Empire against enemies with great mobility (Fig. 1). Field Armies (FAs, comitatenses in Latin) were created and perfected by Emperor Diocletian and Constantine to function as mobile field forces that intercept invading enemy forces before attacks. Although the validity of the implementation of this “defense-in-depth” strategy is still being debated among historians, researchers have been investigating this problem in the past decades as it is closely related to many contemporary problems.
Revelle and Rosing [25] proposed an integer programming model and assumed that in general, a Roman province can be secured by only one FA, but with the support of another, an FA was able to move to an adjacent province and defeat the invading enemy forces there. Their goal was to assign a minimum number of FAs across the Empire to protect it from one attack that could happen with equal probability in any province. As one of several strategies, Revelle and Rosing [25] assigned two FAs at Rome, one FA at Britain and one FA at Asia Minor. Two FAs were assigned to Rome because of its strategical and political significance. Other five provinces with no FAs were secured by one of two FAs at Rome, while FAs at Britain and Asia Minor could not move and only covered local provinces. Of course, assigning two FAs at Iberia as well as Egypt is another optimal strategy, where FAs at both provinces can travel to adjacent provinces and defend them.
The original RDP resembles a well-known facility location problem: the maximal covering location problem (MCLP), where facilities are assigned to a graph so that the expected coverage of demand locations is maximized [24]. We can consider the province under attack as a location of demand and the province with one or two FAs as a location of supply. Suppose that the number of FAs is given in advance, the RDP can be rewritten in a way that FA coverage of other provinces is maximized. For narrative convenience, we use word “provinces” and “locations” interchangeably to refer to the vertices of a graph. However, a maximal coverage in the RDP does not necessarily produce solutions with a “perfect defense”, meaning that whichever province is under attack, there is always an FA that either can move to the province or was stationed at that province before the attack. In fact, Revelle and Rosing [25] considered a perfect defense scenario by modeling the RDP as a set covering facility location problem (SCFLP). Through their calculation, the minimal number of FAs required for a total defense on Fig. 1 is four.
Since Revelle and Rosing [25], few research has concentrated on algorithms that produce optimal strategies for the RDP, nor solving real-world challenges. Yet, gaps still exist between assumption-based models and practical situations. In reality, defenders often have to prepare for multiple simultaneous threats. Nowadays antiterrorism strategies can serve as an example. On Easter April 21, 2019, seven populated locations in Sri Lanka were targeted by terrorist bombings. The coordinated attacks, happened within a short window of seven hours, killed more than 250 citizens from 15 countries. The Sri Lanka attack signifies a challenge facing counterterrorism: attacks that are coordinated to take place at multiple locations within a short period of time, or Complex Coordinated Terrorist Attacks (CCTA). According to the Federal Emergency Management Agency (FEMA), operational coordination and communication is paramount when addressing CCTA. Potential applications of the “defense-in-depth” strategy also extend to other areas such as disaster relief, supply chain disruptions, critical infrastructure resilience, etc. To tackle such real-world challenges, we consider an extension to the original RDP with multiple simultaneously attacks, from the perspective of Integer Programming (IP) and Stochastic Programming (SP) formulations and solution methods.
Section snippets
Literature review
Following Revelle and Rosing [25] and Cockayne et al. [12] studied the RDP from the angle of graph theory and defined key terminologies, such as Roman Domination Function (RDF) and Roman Domination Number (RDN), which have been widely accepted and used in the literature. Let denotes a graph, where is the set of all vertices and the set of all edges. Define a function with a partition of set , where . Let denote the set of all
Definition
In this section, we first discuss, in detail, two existing extensions on the RDP that considered multiple simultaneous attacks and how they can be improved. Then, we introduce new concepts and formally define -SRDP.
Model
In this section, we first propose an IP formulation of the -SRDP. Then, we decompose the problem using Benders decomposition and propose an equivalent two-stage SP formulation. We prove properties of the second stage program and show that from the properties, two algorithms can be derived.
Computational study
In this section, experiments are conducted on randomly generated instances to show model results and performances. A comparison is made between the IP formulation, the revised L-shaped method and the bundled L-shaped method. All models are coded and solved using Gurobi 8.1.0 on an iMac (High Sierra) computer with a 4 Core 2.93 GHz CPU and 8 GB RAM.
Conclusion
In this study, we investigate theoretical aspects of a “defense-in-depth” strategy that optimally protects a graph or network from multiple simultaneous attacks. Specifically, we extend the RDP and propose the -SRDP. We propose an IP formulation of the -SRDP and then decompose it with Benders decomposition, resulting in a two-stage stochastic program. We prove that the second-stage left-hand-side matrix is totally unimodular, so that the corresponding LP relaxation is a perfect formulation.
CRediT authorship contribution statement
Zeyu Liu: Methodology, Software. Xueping Li: Conceptualization, Methodology, Supervision, . Anahita Khojandi: Supervision.
Acknowledgment
This work was supported in part by the National Science Foundation Grant CMMI-1634975.
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