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Resource Minimization for Fire Containment
"... We consider the following model for fire containment. We are given an undirected graph G = (V, E) with a source vertex s where the fire starts. At each time step, the firefighters can save up to k vertices of the graph, while the fire spreads from burning vertices to all their neighbors that have no ..."
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We consider the following model for fire containment. We are given an undirected graph G = (V, E) with a source vertex s where the fire starts. At each time step, the firefighters can save up to k vertices of the graph, while the fire spreads from burning vertices to all their neighbors that have not been saved so far. Our goal is to choose the vertices to be saved at each time step so as to contain the fire. This is a simple mathematical model abstracting the dynamic nature of fire containment and other natural processes, such as, for example, the spread of a perfectly contagious disease and its containment via vaccination. We focus on the Resource Minimization Fire Containment (RMFC) problem, where we are additionally given a subset T ⊆ V of vertices called terminals that need to be protected from fire. The objective is to minimize k the maximum number of vertices to be saved at any time step, so that the fire does not spread to the vertices of T. The problem is hard to approximate up to any factor better than 2 even on trees. We show an O(log ∗ n)approximation LProunding algorithm for RMFC on trees. We also show that an even stronger LP relaxation has an integrality gap of Ω(log ∗ n) on trees. Finally, we consider RMFC on directed layered graphs, and show an O(log n)approximation LProunding algorithm, matching the integrality gap of the LP relaxation. 1
Politician’s Firefighting
"... Abstract. Firefighting is a combinatorial optimization problem on graphs that models the problem of determining the optimal strategy to contain a fire and save as much (trees, houses, etc.) from the fire as possible. We study a new version of firefighting, which we call Politician’s Firefighting and ..."
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Abstract. Firefighting is a combinatorial optimization problem on graphs that models the problem of determining the optimal strategy to contain a fire and save as much (trees, houses, etc.) from the fire as possible. We study a new version of firefighting, which we call Politician’s Firefighting and which exhibits more locality than the classical onefirefighter version. We prove that this locality allows us to develop an O(bn)time algorithm for this problem on trees, where b is the number of nodes initially on fire. We further prove that Politician’s Firefighting is NPhard on planar graphs of degree at least 5, and we present an O(m+k 2.5 4 k)time algorithm for this problem on general graphs, where k is the number of nodes that burn using the optimal strategy, thereby proving that it is fixedparameter tractable. We present experimental results that show that our algorithm’s search tree size is in practice much smaller than the worstcase bound of 4 k. 1
Computer science and decision theory
 Annals of Operations Research
"... This paper reviews applications in computer science that decision theorists have addressed for years, discusses the requirements posed by these applications that place great strain on decision theory/social science methods, and explores applications in the social and decision sciences of newer decis ..."
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This paper reviews applications in computer science that decision theorists have addressed for years, discusses the requirements posed by these applications that place great strain on decision theory/social science methods, and explores applications in the social and decision sciences of newer decisiontheoretic methods developed with computer science applications in mind. The paper deals with the relation between computer science and decisiontheoretic methods of consensus, with the relation between computer science and game theory and decisions, and with “algorithmic decision theory.” 1
Greedy Algorithms in Economic Epidemiology
 DIMACS SERIES IN DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
"... Economic issues are central to the control of disease because of the limited funds available for public health everywhere in the world, even in the wealthiest nations. These economic issues are closely related to issues of individual human behavior, as well as to fundamental disease processes and t ..."
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Economic issues are central to the control of disease because of the limited funds available for public health everywhere in the world, even in the wealthiest nations. These economic issues are closely related to issues of individual human behavior, as well as to fundamental disease processes and their relation to the environment. While mathematical formulation of epidemiological processes is an old discipline, combining such formulations with economic, behavioral, and environmental formalisms is relatively new, and has come to define the field of “Economic Epidemiology.” Many problems in Economic Epidemiology can be formulated as optimization problems. The simplest approach to solving such a problem is often a greedy algorithm, one that always chooses the best available (cheapest, highest rated,...) alternative at each step. We review some classical operations research problems arising in Economic Epidemiology for which the greedy algorithm in fact gives an optimal solution, and others for which it can be guaranteed to be reasonably close. We then present two examples from our own work. Examples will be chosen from: assigning workers to health care tasks; choosing medical supplies to maximize value and minimize cost; locating a health care facility so as to minimize the travel times of users; and reopening flooded roads to allow the passage of emergency vehicles. Other examples will include optimal strategies for vaccination given a limited supply; and optimal strategies for sequencing medical tests or public health interventions in order to minimize costs and maximize success.
Irreversible kthreshold and majority conversion processes on complete multipartite graphs and graph products
"... In graph theoretical models of the spread of disease through populations, the spread of opinion through social networks, and the spread of faults through distributed computer networks, vertices are in two states, either black or white, and these states are dynamically updated at discrete time steps ..."
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In graph theoretical models of the spread of disease through populations, the spread of opinion through social networks, and the spread of faults through distributed computer networks, vertices are in two states, either black or white, and these states are dynamically updated at discrete time steps according to the rules of the particular conversion process used in the model. This paper considers the irreversible kthreshold and majority conversion processes. In an irreversible kthreshold (resp., majority) conversion process, a vertex is permanently colored black in a certain time period if at least k (resp., at least half) of its neighbors were black in the previous time period. A kconversion set (resp., dynamic monopoly) is a set of vertices which, if initially colored black, will result in all vertices eventually being colored black under a kthreshold (resp., majority) conversion process. We answer several open problems by presenting bounds and some exact values of the minimum number of vertices in kconversion sets and dynamic monopolies of complete multipartite graphs, as well as of Cartesian and tensor products of two graphs.