It is well known that fighting a fire is a hard task. The Firefighter problem asks how to optimally deploy firefighters to defend the vertices of a graph from a fire. This problem is NP-Complete on all but a few classes of graphs. Thankfully, firefighters do not have to work alone, and are often aided by the efforts of good natured civilians who slow the spread of a fire by maintaining firebreaks when they are able. We will show that this help, although well-intentioned, unfortunately makes the optimal deployment of firefighters an even harder problem. To model this scenario we introduce the Temporal Firefighter problem, an extension of Firefighter to temporal graphs. We show that Temporal Firefighter is also NP-Complete, and remains so on all but one of the underlying classes of graphs on which Firefighter is known to have polynomial time solutions. This motivates us to explore making use of the temporal structure of the graph in our search for tractability, and we conclude by presenting an FPT algorithm for Temporal Firefighter with respect to the temporal graph parameter vertex-interval-membership-width.
In firefighting and other emergency operations, decisions made under pressure carry profound ethical weight and can significantly impact incident outcomes and firefighter safety. Traditional training methods, while foundational, often fall short in adequately preparing firefighters for the complex ethical dilemmas and value conflicts inherent in chaotic emergency environments. This paper proposes a conceptual framework for enhancing firefighter training by systematically integrating departmental values into simulation-based training. This approach fosters deeper value internalisation and improves value-driven decision-making under pressure. Furthermore, the underlying tools can also be leveraged to evaluate and refine departmental operational protocols for better alignment with preferred values.
PPE (Personal Protective Equipment) has allowed firefighters to perform their everyday tasks without getting harmed since the mid 1800s. Now, the advancement of technology has given rise to the improvements of PPE. PPE can now include sensors to detect any number of environmental hazards (chemical, biological, temperature etc.). As the GT class of CS3750, we have decided to create a version of an interface design sensor that will help firefighters in two ways: navigation and communication. In order to augment a firefighter display when they are within a building, we chose to augment their SCBA (self-contained breathing apparatus). The gas mask will include a small screen that displays vital information directly towards the firefighter without need of any other support. We used the Google Glass to display vital information directly towards the eye in a minimalistic manner, while also augmenting that by adding LED lights to simulate someone calling their name or other auditory signals.While our prototype focuses on two main components of a firefighters search and rescue in a building, both of them combine to augment a firefighters display when searching throughout a building to help
This paper presents the Firefighter Optimization (FFO) algorithm as a new hybrid metaheuristic for optimization problems. This algorithm stems inspiration from the collaborative strategies often deployed by firefighters in firefighting activities. To evaluate the performance of FFO, extensive experiments were conducted, wherein the FFO was examined against 13 commonly used optimization algorithms, namely, the Ant Colony Optimization (ACO), Bat Algorithm (BA), Biogeography-Based Optimization (BBO), Flower Pollination Algorithm (FPA), Genetic Algorithm (GA), Grey Wolf Optimizer (GWO), Harmony Search (HS), Particle Swarm Optimization (PSO), Simulated Annealing (SA), Tabu Search (TS), and Whale Optimization Algorithm (WOA), and across 24 benchmark functions of various dimensions and complexities. The results demonstrate that FFO achieves comparative performance and, in some scenarios, outperforms commonly adopted optimization algorithms in terms of the obtained fitness, time taken for exaction, and research space covered per unit of time.
Applications of drones in emergency response, like firefighting, have been promoted in the past decade. As the autonomy of drones continues to improve, the ways in which they are integrated into firefighting teams and their impact on crews are changing. This demands more understanding of how firefighters perceive and interact with autonomous drones. This paper presents a drone-based system for emergency operations with which firefighters can interact through sound, lights, and a graphical user interface. We use interviews with stakeholders collected in two field trials to explore their perceptions of the interaction and collaboration with drones. Our result shows that firefighters perceived visual interaction as adequate. However, for audio instructions and interfaces, information overload emerges as an essential problem. The potential impact of drones on current work configurations may involve shifting the position of humans closer to supervisory decision-makers and changing the training structure and content.
The containment game is a full information game for two players, initialised with a set of occupied vertices in an infinite connected graph $G$. On the $t$-th turn, the first player, called Spreader, extends the occupied set to $g(t)$ adjacent vertices, and then the second player, called Container, removes $q$ unoccupied vertices from the graph. If the spreading process continues perpetually -- Spreader wins, and otherwise -- Container wins. For $g=\infty$ this game reduces to a solitaire game for Container, known as the Firefighter Problem. On $\mathbb{Z}^2$, for $q=1/k$ and $g\equiv 1$ it is equivalent to Conway's Angel Problem. We introduce the game, and writing $q(G,g)$ for the set of $q$ values for which Container wins against a given $g(t)$, we study the minimal asymptotics of $g(t)$ such that $q(G,g)=q(G,\infty)$, i.e. for which defeating Spreader is as hard as winning the Firefighter Problem solitaire. We show, by providing explicit winning strategies, a sub-linear upper bound $g(t)=O(t^{6/7})$ and a lower bound of $g(t)=Ω(t^{1/2})$.
Unit disk graphs are the set of graphs which represent the intersection of disk graphs and interval graphs. These graphs are of great importance due to their structural similarity with wireless communication networks. Firefighter problem on unit disk graph is interesting as it models the virus spreading in an wireless network and asks for a solution to stop it. In this paper, we consider the MIN-BUDGET firefighter problem where the goal is to determine the minimum number of firefighters required and the nodes to place them at each time instant to save a given set of vertices of a given graph and a fire breakout node. We show that, the MIN-BUDGET firefighter problem in a unit disk graph is NP-Hard. We also present a constant factor approximation algorithm.
The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.
We consider the problem of firefighting to save a critical subset of nodes. The firefighting game is a turn-based game played on a graph, where the fire spreads to vertices in a breadth-first manner from a source, and firefighters can be placed on yet unburnt vertices on alternate rounds to block the fire. In this work, we consider the problem of saving a critical subset of nodes from catching fire, given a total budget on the number of firefighters. We show that the problem is para-NP-hard when parameterized by the size of the critical set. We also show that it is fixed-parameter tractable on general graphs when parameterized by the number of firefighters. We also demonstrate improved running times on trees and establish that the problem is unlikely to admit a polynomial kernelization (even when restricted to trees). Our work is the first to exploit the connection between the firefighting problem and the notions of important separators and tight separator sequences. Finally, we consider the spreading model of the firefighting game, a closely related problem, and show that the problem of saving a critical set parameterized by the number of firefighters is W[2]-hard, which contrasts
In this paper we study the complexity of the firefighter problem and related problems on trees when more than one firefighter is available at each time step, and answer several open questions of Finbow and MacGillivray 2009. More precisely, when $b \geq 2$ firefighters are allowed at each time step, the problem is NP-complete for trees of maximum degree $b+2$ and polynomial-time solvable for trees of maximum degree $b+2$ when the fire breaks out at a vertex of degree at most $b+1$. Moreover we present a polynomial-time algorithm for a subclass of trees, namely $k$-caterpillars.
This paper introduces Galactic Bioware's Life Support System, a semi-closed-circuit breathing apparatus designed for integration into a positive-pressure firefighting suit and governed by an AI control system. The breathing loop incorporates a soda lime CO2 scrubber, a silica gel dehumidifier, and pure O2 replenishment with finite consumables. One-way exhaust valves maintain positive pressure while creating a semi-closed system in which outward venting gradually depletes the gas inventory. Part I develops the physicochemical foundations from first principles, including state-consistent thermochemistry, stoichiometric capacity limits, adsorption isotherms, and oxygen-management constraints arising from both fire safety and toxicity. Part II introduces an AI control architecture that fuses three sensor tiers, external environmental sensing, internal suit atmosphere sensing (with triple-redundant O2 cells and median voting), and firefighter biometrics. The controller combines receding-horizon model-predictive control (MPC) with a learned metabolic model and a reinforcement learning (RL) policy advisor, with all candidate actuator commands passing through a final control-barrier-functi
Allocation of personnel and material resources is highly sensible in the case of firefighter interventions. This allocation relies on simulations to experiment with various scenarios. The main objective of this allocation is the global optimization of the firefighters response. Data generation is then mandatory to study various scenarios In this study, we propose to compare different data generation methods. Methods such as Random Sampling, Tabular Variational Autoencoders, standard Generative Adversarial Networks, Conditional Tabular Generative Adversarial Networks and Diffusion Probabilistic Models are examined to ascertain their efficacy in capturing the intricacies of firefighter interventions. Traditional evaluation metrics often fall short in capturing the nuanced requirements of synthetic datasets for real-world scenarios. To address this gap, an evaluation of synthetic data quality is conducted using a combination of domain-specific metrics tailored to the firefighting domain and standard measures such as the Wasserstein distance. Domain-specific metrics include response time distribution, spatial-temporal distribution of interventions, and accidents representation. These m
The firefighter problem with $k$ firefighters on an infinite graph $G$ is an iterative graph process, defined as follows: Suppose a fire breaks out at a given vertex $v\in V(G)$ on Turn 1. On each subsequent even turn, $k$ firefighters protect $k$ vertices that are not on fire, and on each subsequent odd turn, any vertex that is on fire spreads the fire to all adjacent unprotected vertices. The firefighters' goal is to eventually stop the spread of the fire. If there exists a strategy for $k$ firefighters to eventually stop the spread of the fire, then we say $G$ is $k$-containable. We consider the firefighter problem on the hexagonal grid, which is the graph whose vertices and edges are exactly the vertices and edges of a regular hexagonal tiling of the plane. It is not known if the hexagonal grid is $1$-containable. In arXiv:1305.7076 [math.CO], it was shown that if the firefighters have one firefighter per turn and one extra firefighter on two turns, the firefighters can contain the fire. We improve on this result by showing that even with only one extra firefighter on one turn, the firefighters can still contain the fire. In addition, we explore $k$-containability for birth seq
This study presents the first systematic investigation of the dynamics of individual water droplets in the context of airtanker firefighting. While previous work has focused on ground-deposition patterns measured in standardized field tests, the droplet-scale mechanisms governing evaporation and transport have remained largely unexplored. A tailored model of the coupled momentum, heat, and mass transfer of an isolated water droplet in ambient air is proposed and applied to examine the evolution of droplets under a wide range of atmospheric conditions. The results demonstrate that droplet size governs the effectiveness of water delivery, the release height emerges as the dominant operational parameter, and relative humidity is the key atmospheric property. Increasing the release height lengthens the flight time and increases evaporative losses, while low relative humidity accelerates evaporation, particularly for droplets smaller than one millimeter. Only droplets within a narrow range of initial radii, $150\,μ\mathrm{m} \lesssim r_{\mathrm{d},0} \lesssim 3\,\mathrm{mm}$, are able to reach the ground following an airtanker release, with smaller droplets fully evaporating during thei
It is known that the online firefighting is 2-competitive on trees (Coupechoux et al. 2019), which suggests that the problem is relatively easy on trees. We extend the study to graphs containing cycles. We first show that the presence of cycles gives a strong advantage to the adversary: cycles create situations where the algorithm and the optimal solution operate on different game states, and the adversary can exploit the uncertainty in the firefighter sequence to trap the algorithm. Specifically, we prove that even on a tadpole graph (a cycle with a tail path), no deterministic online algorithm achieves a competitive ratio better than $Ω(\sqrt{n})$, where n is the number of vertices. We then propose an $O(\sqrt{n})$-competitive algorithm for 1-almost trees, which contain at most one cycle and generalize tadpole graphs. We further generalize this algorithm to cactus graphs, in which multiple cycles may appear, but no two share more than one vertex, and show that the online firefighting problem on cactus graphs remains $O(\sqrt{n})$-competitive. Finally, since cactus graphs have treewidth at most 2, we study a variant where firefighters are released in pairs, that is, each round an
In the Firefighter problem, introduced by Hartnell in 1995, a fire spreads through a graph while a player chooses which vertices to protect in order to contain it. In this paper, we focus on the case of trees and we consider as well the Fractional Firefighter game where the amount of protection allocated to a vertex lies between 0 and 1. While most of the work in this area deals with a constant amount of firefighters available at each turn, we consider three research questions which arise when including the sequence of firefighters as part of the instance. We first introduce the online version of both Firefighter and Fractional Firefighter, in which the number of firefighters available at each turn is revealed over time. We show that a greedy algorithm on finite trees is 1/2-competitive for both online versions, which generalises a result previously known for special cases of Firefighter. We also show that the optimal competitive ratio of online Firefighter ranges between 1/2 and the inverse of the golden ratio. Next, given two firefighter sequences, we discuss sufficient conditions for the existence of an infinite tree that separates them, in the sense that the fire can be contain
Drone fleets equipped with onboard cameras, computer vision, and Deep Neural Network (DNN) models present a powerful paradigm for real-time spatio-temporal decision-making. In wildfire response, such drones play a pivotal role in monitoring fire dynamics, supporting firefighter coordination, and facilitating safe evacuation. In this paper, we introduce AeroResQ, an edge-accelerated UAV framework designed for scalable, resilient, and collaborative escape route planning during wildfire scenarios. AeroResQ adopts a multi-layer orchestration architecture comprising service drones (SDs) and coordinator drones (CDs), each performing specialized roles. SDs survey fire-affected areas, detect stranded individuals using onboard edge accelerators running fire detection and human pose identification DNN models, and issue requests for assistance. CDs, equipped with lightweight data stores such as Apache IoTDB, dynamically generate optimal ground escape routes and monitor firefighter movements along these routes. The framework proposes a collaborative path-planning approach based on a weighted A* search algorithm, where CDs compute context-aware escape paths. AeroResQ further incorporates intell
Fires in industrial facilities pose special challenges to firefighters, e.g., due to the sheer size and scale of the buildings. The resulting visual obstructions impair firefighting accuracy, further compounded by inaccurate assessments of the fire's location. Such imprecision simultaneously increases the overall damage and prolongs the fire-brigades operation unnecessarily. We propose an automated assistance system for firefighting using a motorized fire monitor on a turntable ladder with aerial support from an unmanned aerial vehicle (UAV). The UAV flies autonomously within an obstacle-free flight funnel derived from geodata, detecting and localizing heat sources. An operator supervises the operation on a handheld controller and selects a fire target in reach. After the selection, the UAV automatically plans and traverses between two triangulation poses for continued fire localization. Simultaneously, our system steers the fire monitor to ensure the water jet reaches the detected heat source. In preliminary tests, our assistance system successfully localized multiple heat sources and directed a water jet towards the fires.
In the classic version of the game of firefighter, on the first turn a fire breaks out on a vertex in a graph $G$ and then $b$ firefighters protect $b$ vertices. On each subsequent turn, the fire spreads to the collective unburned neighbourhood of all the burning vertices and the firefighters again protect $b$ vertices. Once a vertex has been burned or protected it remains that way for the rest of the game. In \textit{distance-restricted firefighting} the firefighters' movement is restricted so they can only move up to some fixed distance $d$ and they may or may not be permitted to move through burning vertices. In this paper we establish the NP-completeness of the distance-restricted versions of {\sc $b$-Firefighter} and present an integer program for computing the exact value. We also discuss some interesting properties of the \textit{Expected Damage} function.
We consider a pursuit-evasion game that describes the process of extinguishing a fire burning on the nodes of an undirected graph. We denote the minimum number of firefighters required by ffn(G) and provide almost sharp bounds to this graph parameter for complete binary trees. We show that deciding whether ffn(G) <= m for given G and m is NP-hard. Furthermore, we show that shortest strategies can have superpolynomial length, leaving open whether the problem is in NP. We provide a construction that allows for transferring these results to a well-established Cops and Robbers variant called the "Hunter and Rabbit game".