Publication: Cómo frenar una propagación: Análisis de complejidad y resolución
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Publication Date
2021
Advisors (or tutors)
Rubio Diez, Fernando
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Abstract
Las propagaciones en todas sus variantes constituyen una parte fundamental de la existencia humana. Las enfermedades, un virus informático o incluso la propagación de los impulsos nerviosos suponen ejemplos muy variados que reflejan la importancia del estudio de las mismas, para entender su naturaleza y, en muchos casos, cómo erradicarlas. Este problema ha concentrado el esfuerzo de muchos profesionales de todos los ámbitos, como la medicina o la psicología. Otros estudios previos han investigado cómo las propagaciones avanzan por un entorno, llevando a cabo un acercamiento al problema desde un punto de vista mucho más estático, pues no adaptan su estrategia para contener la propagación de manera dinámica como nosotros pretendemos. A pesar de esto, los modelos que se basan en predecir la forma en la que una propagación avanza suelen fallar debido a la gran cantidad de variables en consideración y al desconocimiento inicial cuando una propagación emerge.
Este estudio nos permitirá adentrarnos de una forma genérica, teórica y práctica en el funcionamiento de las propagaciones; qué podemos hacer contra ellas y cuál es la dificultad de frenarlas. A lo largo de este escrito profundizaremos en las propagaciones desde una perspectiva distinta, la teoría de complejidad. Probaremos que el problema de encontrar una estrategia óptima para lograr frenar una propagación es PSPACE-completo bajo determinadas condiciones. La intratabilidad de este problema hace necesario el uso de métodos heurísticos para solucionarlo de manera subóptima, en nuestro caso, un algoritmo minimax con poda alfabeta. De esta manera, crearemos un modelo desde cero que nos servirá para representar el arranque de una propagación y la inmediata intervención del apaciguador, figura que simulará la acción humana en el intento de frenar la misma. Nos apoyaremos en ejemplos representativos que nos permitirán analizar las fortalezas y debilidades del modelo y nos ayudarán a entender de manera sencilla el funcionamiento del algoritmo minimax aplicado a las propagaciones. Asimismo, este trabajo tiene la finalidad de suponer una mejora social en determinados sectores pues tiene aplicabilidad real, brindando una herramienta que permita adelantarse a aquellas complicaciones que podría generar cualquier tipo de propagación. Para ello, presentaremos un caso de estudio donde basamos nuestro problema en datos reales, extraídos de la propagación de la COVID-19 en España, como son el Presupuesto de Sanidad Pública para cada una de las Comunidades Autónomas o el porcentaje de Movilidad interterritorial en Marzo de 2020.
Propagations in all their variants are a fundamental part of human existence. Diseases, computer viruses, or even nerve impulses are very varied examples that reflect the importance of studying them to understand their nature and, in some cases, how to eradicatethem. This problem has concentrated the efforts of many professionals from all fields, such as medicine or psychology. Other previous works have researched on how a propagation spreads through an environment, addressing the problem from a much more static perspective, as they do not adapt their strategy to stop the propagation dynamically as we intend. However, those models that are based on predicting how a propagation progresses often fail, the reason being the large number of variables under consideration and the initial lack of knowledge when a propagation emerges. This study will allow us to enter into a general, theoretical, and practical approach about how propagations work; what can we do against them, and how complex is to stop it. Throughout this paper, we will deepen in propagations from a new approach, the complexity theory. We will prove that the problem of finding an optimal strategy for stopping a propagation is PSPACE-complete under some standard assumptions. The intractability of this problem makes it necessary to use heuristic methods to solve it in a suboptimal way, in our case, a minimax algorithm with alpha-beta pruning. In this way, we will create a model from scratch that will serve to represent the beginning of propagation and the immediate intervention of the appeaser, a figure that will simulate the human action attempting to stop it. We will use representative examples that will allow us to analyse the strengths and weaknesses of the model and will help us to understand in a simple way how the minimax algorithm applied to propagations works. Likewise, this work is intended to bring about a social improvement in certain sectors as it has real applicability, providing a tool that allows us to anticipate those complications that could be generated by any type of propagation. To this end, we will present a case study where we base our problem on real data, extracted from the COVID-19 propagation in Spain, such as the Public Health budget for each region or the percentage of interterritorial mobility from March 2020.
Propagations in all their variants are a fundamental part of human existence. Diseases, computer viruses, or even nerve impulses are very varied examples that reflect the importance of studying them to understand their nature and, in some cases, how to eradicatethem. This problem has concentrated the efforts of many professionals from all fields, such as medicine or psychology. Other previous works have researched on how a propagation spreads through an environment, addressing the problem from a much more static perspective, as they do not adapt their strategy to stop the propagation dynamically as we intend. However, those models that are based on predicting how a propagation progresses often fail, the reason being the large number of variables under consideration and the initial lack of knowledge when a propagation emerges. This study will allow us to enter into a general, theoretical, and practical approach about how propagations work; what can we do against them, and how complex is to stop it. Throughout this paper, we will deepen in propagations from a new approach, the complexity theory. We will prove that the problem of finding an optimal strategy for stopping a propagation is PSPACE-complete under some standard assumptions. The intractability of this problem makes it necessary to use heuristic methods to solve it in a suboptimal way, in our case, a minimax algorithm with alpha-beta pruning. In this way, we will create a model from scratch that will serve to represent the beginning of propagation and the immediate intervention of the appeaser, a figure that will simulate the human action attempting to stop it. We will use representative examples that will allow us to analyse the strengths and weaknesses of the model and will help us to understand in a simple way how the minimax algorithm applied to propagations works. Likewise, this work is intended to bring about a social improvement in certain sectors as it has real applicability, providing a tool that allows us to anticipate those complications that could be generated by any type of propagation. To this end, we will present a case study where we base our problem on real data, extracted from the COVID-19 propagation in Spain, such as the Public Health budget for each region or the percentage of interterritorial mobility from March 2020.
Description
Trabajo de Fin de Doble Grado en Ingeniería Informática y Matemáticas, Facultad de Informática UCM, Departamento de Sistemas Informáticos y Computación, Curso 2020/2021.