A matheuristic for the firefighter problem on graphs

In this paper, we describe a computational study conducted on The Firefighter Problem (FFP ), which models fire spreading and containment in a graph. Once the fire breaks out on a set of vertices, the goal is to save as many vertices as possible with limited resources. Practical applications of the FFP occur in areas such as disease control and network security. The FFP is NP‐hard and heuristics have been proposed earlier. Our main contributions include improvements to an existing integer linear programming formulation that led to an average speedup of two to compute exact solutions. Moreover, we developed a novel matheuristic, a technique based on the interoperation between metaheuristics and mathematical programming. We performed extensive experiments on public benchmarks both for parameter tuning and for comparison of our results with those from the literature. A rigorous statistical analysis shows that our new matheuristic outperforms the existing approaches.     

The firefighter problem: Empirical results on random graphs

The firefighter problem is a deterministic discrete-time model for the spread and containment of fire on a graph. Once the fire breaks out at a set of vertices, the goal addressed in this work is to save as many vertices as possible from burning. Although the problem finds applications in various real-world problems, such as the spread of diseases or hoaxes contention in communication networks, this problem has not been addressed from a practical point of view so far, in the sense of finding a good strategy for the general case. In this work, we develop and compare several integer linear programming techniques and heuristic methods. Random graphs are used for the purpose of comparison. The obtained results shed some light on the challenges for computational tools as caused by graph topology, graph size, and the number of firefighters per iteration, when looking for the best strategy for an a priori unknown graph.     

The online graph bandwidth problem

The online graph bandwidth problem is defined, and we present an online algorithm that always outputs a (((2k − 1)n + 1)/2k)-bandwidth function for any n-vertex graph with bandwidth k. A lower bound of (k/(k + 1))n − 2 is shown for any such algorithm. Two other protocols for online problems are given, and we prove lower bounds for the bandwidth problem under both of these alternative protocols.     

求解GCP问题的ILSBR算法

图着色问题（GCP,Graph Coloring Problem）是经典的NP-Hard组合优化问题之一。长期以来,人们一直在寻求快速、高效的启发式算法,以便在合理的计算时间内解决大规模问题。由于对规模较大的问题,目前的启发式算法尚不能在较短的时间内给出高质量的解,因此提出了一种基于全局最优解和局部最优解关系的ILS算法（ILSBR）。该算法的基本原理是通过对GCP问题的局部最优解和全局最优解之间关系的分析,发现对局部最优解的简单的相交操作能以很高的概率得到全局最优解的部分解。利用这些部分解构造一种新的扰动策略（RLG重着色）,并将其应用到传统的ILS算法中。在DIMACS标准集中,典型实例上的实验结果表明,采用RBILS算法在求解质量不变的情况下,求解速度上与目前的已知算法相比有较大的改进。     

The optimal information rate for graph access structures of nine participants

The information rate is an important metric of the performance of a secret-sharing scheme. In this paper we consider 272 non-isomorphic connected graph access structures with nine vertices and eight or nine edges, and either determine or bound the optimal information rate in each case. We obtain exact values for the optimal information rate for 231 cases and present a method that is able to derive information-theoretical upper bounds on the optimal information rate. Moreover, we apply some of the constructions to determine lower bounds on the information rate. Regarding information rate, we conclude with a full listing of the known optimal information rate (or bounds on the optimal information rate) for all 272 graphs access structures of nine participants.     

A parametric filtering algorithm for the graph isomorphism problem

We introduce a new filtering algorithm, called IDL(d)-filtering, for a global constraint dedicated to the graph isomorphism problem—the goal of which is to decide if two given graphs have an identical structure. The basic idea of IDL(d)-filtering is to label every vertex with respect to its relationships with other vertices around it in the graph, and to use these labels to filter domains by removing values that have different labels. IDL(d)-filtering is parameterized by a positive integer value d which gives a limit on the distance between a vertex to be labelled and the set of vertices considered to build its label. We experimentally compare different instantiations of IDL(d)-filtering with state-of-the-art dedicated algorithms and show that IDL(d)-filtering is more efficient on regular sparse graphs and competitive on other kinds of graphs.     

A problem of tree graph

Because of the widespread applications of tree and treee graph in computer science,we are interested in studying the reee graph.M.Farber,B.Richter and H.Shang in [1] showed that the graph τ2(G)is 2-edge-connected as |V(G)）≥3,at the same time,we will show the best lower bounds about vertex number and minimum degree of graph τ2(G）.     

A hierarchical parallel genetic approach for the graph coloring problem

Graph Coloring Problems (GCPs) are constraint optimization problems with various applications including time tabling and frequency allocation. The GCP consists in finding the minimum number of colors for coloring the graph vertices such that adjacent vertices have distinct colors. We propose a hierarchical approach based on Parallel Genetic Algorithms (PGAs) to solve the GCP. We call this new approach Hierarchical PGAs (HPGAs). In addition, we have developed a new operator designed to improve PGAs when solving constraint optimization problems in general and GCPs in particular. We call this new operator Genetic Modification (GM). Using the properties of variables and their relations, GM generates good individuals at each iteration and inserts them into the PGA population in the hope of reaching the optimal solution sooner. In the case of the GCP, the GM operator is based on a novel Variable Ordering Algorithm (VOA) that we propose. Together with the new crossover and the estimator of the initial solution we have developed, GM allows our solving approach to converge towards the optimal solution sooner than the well known methods for solving the GCP, even for hard instances. This was indeed clearly demonstrated by the experiments we conducted on the GCP instances taken from the well known DIMACS website.     

An exact algorithm with learning for the graph coloring problem

Given an undirected graph G=(V,E), the Graph Coloring Problem (GCP) consists in assigning a color to each vertex of the graph G in such a way that any two adjacent vertices are assigned different colors, and the number of different colors used is minimized. State-of-the-art algorithms generally deal with the explicit constraints in GCP: any two adjacent vertices should be assigned different colors, but do not specially deal with the implicit constraints between non-adjacent vertices implied by the explicit constraints. In this paper, we propose an exact algorithm with learning for GCP which exploits the implicit constraints using propositional logic. Our algorithm is compared with several exact algorithms among the best in the literature. The experimental results show that our algorithm outperforms other algorithms on many instances. Specifically, our algorithm allows to close the open DIMACS instance 4-Fullins_5.     

On the feedback vertex set problem for a planar graph

An algorithm solving the feedback-vertex-set problem for planar digraphs is described. In particular, planar graphs with a certain additional condition are considered as they arise from solving systems of linear equations obtained from convection-dominated flow problems. The proposed algorithm requires a computational work linear in the size of the graph. Furthermore, a side-product is a decomposition of the graph into subsets that are of interest for block-Gauß-Seidel smoothers in multi-grid methods.     

A neural-network algorithm for a graph layout problem

We present a neural-network algorithm for minimizing edge crossings in drawings of nonplanar graphs. This is an important subproblem encountered in graph layout. The algorithm finds either the minimum number of crossings or an approximation thereof and also provides a linear embedding realizing the number of crossings found. The parallel time complexity of the algorithm is O(1) for a neural network with n(2) processing elements, where n is the number of vertices of the graph. We present results from testing a sequential simulator of the algorithm on a set of nonplanar graphs and compare its performance with the heuristic of Nicholson.     

MTPSO algorithm for solving planar graph coloring problem

In this paper, we proposed a modified turbulent particle swarm optimization (named MTPSO) model for solving planar graph coloring problem based on particle swarm optimization. The proposed model is consisting of the walking one strategy, assessment strategy and turbulent strategy. The proposed MTPSO model can solve the planar graph coloring problem using four-colors more efficiently and accurately. Compared to the results shown in Cui et al. (2008), not only the experimental results of the proposed model can get smaller average iterations but can get higher correction coloring rate when the number of nodes is greater than 30.     

The ageing Australian firefighter: an argument for age-based recruitment and fitness standards for urban fire services

Currently, there is no enforcement of physical standards within Australian fire services post-recruitment, possibly leading to inappropriate fitness and body composition. This study evaluated the impacts of ageing on physical standards of Australian firefighters. Seventy-three firefighters from three different 10-year age groups [25–34 years (n = 27), 35–44 years (n = 27), 45–54 years (n = 19)] volunteered for physical testing using dual-energy X-ray analysis and existing fitness tests used for recruitment by an Australian fire service. Older (45–54 years) participants demonstrated significantly poorer physical standards compared with younger participants including cardiovascular fitness (p < 0.05), strength (p = 0.001) and simulated operational power testing tasks (p < 0.001). Age-related body composition changes were also observed independent of body mass index. Minimum recruitment standards and fitness programs need to account for age-related declines in physical capabilities to ensure that the minimum standard is maintained regardless of age.

Practitioner Summary: Using dual-energy X-ray analysis and established fitness testing protocols, this study aimed to gain an appreciation of the current standards of body composition and fitness of Australian firefighters and the effects of ageing on their physical abilities post-recruitment. The study demonstrated a significant decline in physical standards due to age.     

Weighted nearest neighbor algorithms for the graph exploration problem on cycles

In the graph exploration problem, a searcher explores the whole set of nodes of an unknown graph. We assume that all the unknown graphs are undirected and connected. The searcher is not aware of the existence of an edge until he/she visits one of its endpoints. The searcher's task is to visit all the nodes and go back to the starting node by traveling a tour as short as possible. One of the simplest strategies is the nearest neighbor algorithm (NN), which always chooses the unvisited node nearest to the searcher's current position. The weighted NN (WNN) is an extension of NN, which chooses the next node to visit by using the weighted distance. It is known that WNN with weight 3 is 16-competitive for planar graphs. In this paper we prove that NN achieves the competitive ratio of 1.5 for cycles. In addition, we show that the analysis for the competitive ratio of NN is tight by providing an instance for which the bound of 1.5 is attained, and NN is the best for cycles among WNN with all possible weights. Furthermore, we prove that no online algorithm to explore cycles is better than 1.25-competitive.     

Using an interval graph approach to efficiently solve the housing benefit data retrieval problem

Rappos and Thompson use a set covering formulation and a commercial software package to solve the problem of trying to minimize the number of data sets that have to be read in retrieving all new housing benefit (HB) data entries for a fixed period of time. In this paper, we show that determining the minimum number of data sets that have to be read in retrieving all new HB data entries for a fixed period of time can be solved by finding a minimum size clique cover for an interval graph. Since it is well‐known that a greedy algorithm finds a guaranteed minimum size clique cover for an interval graph, this approach will be more efficient than a set covering approach. Finally, it is obvious that this interval graph formulation and greedy algorithm solution approach is applicable to other data retrieval problems.     

Characterization of graph properties for improved Pareto fronts using heuristics and EA for bi-objective graph coloring problem

Bi-objective graph coloring problem (BOGCP) is a generalized version in which the number of colors used to color the vertices of a graph and the corresponding penalty which incurs due to coloring the end-points of an edge with the same color are simultaneously minimized. In this paper, we have analyzed the graph density, the interconnection between high degree nodes of a graph, the rank exponent of the standard benchmark input graph instances and observed that the characterization of graph instances affects on the behavioral quality of the solution sets generated by existing heuristics across the entire range of the obtained Pareto fronts. We have used multi-objective evolutionary algorithm (MOEA) to obtain improved quality solution sets with the problem specific knowledge as well as with the embedded heuristics knowledge. To establish this fact for BOGCP, hybridization approach is used to construct recombination operators and mutation operators and it is observed from empirical results that the embedded problem specific knowledge in evolutionary operators helps to improve the quality of solution sets across the entire Pareto front; the nature of problem specific knowledge differentiates the quality of solution sets.     

The influence of firefighter boot type on balance

A firefighter's boots play a critical role in working effectiveness and personal safety. OSHA 1910.156 contains standards for personal protective equipment of fire brigades. Firefighters use either rubber or leather boots that meet these requirements. The purpose of the study was to examine the differences in balance in professional firefighters wearing rubber and leather boots when participating in a fire simulation activity. Twelve professional firefighters performed 2 sets of a three-minute simulated firefighter stair climb wearing a 50 lb weighted vest to simulate their typical personal protective equipment and two 5.68 kg weights on the shoulders to simulate the weight of a high-rise pack (hose bundle). On each condition day (leather, rubber) the firefighter conducted a balance assessment. Following the initial balance protocol, the firefighter conducted a Simulated Firefighter Stair Climb for 3 min at a rate of 60 steps per/min. At the completion of the stair climb, the firefighter repeated the balance procedure. Following a 3-minute rest period, the complete procedure (balance, stair climb) was repeated. A total of 3 balance procedures and 2 stair climbs were completed. Significant differences were found in sway velocity between the pre and post test measures and among the two different boots. These results suggest that the rubber boots elicit greater postural instability. These findings provide practical information on work practices and PPE usage decisions.Relevance to industryIndustry standards dictate the protective variables of boots used by fire brigades, but do not consider the influence on gait and balance. This study provides evidence that the rubber boots used by firefighters may impair specific balance parameters which are critical for firefighter safety.