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Transcript of Optimization on Network and Graph
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Э.Майника АЛГОРИТМЫ ОПТИМИЗАЦИИ НА СЕТЯХ И ГРАФАХ
М.: Мир, 1981, 324 стр. Предисловие редактора перевода 5 Предисловие 7 Глава 1. Введение в теорию графов и сетей 9 1.1. Вводные замечания 9 1.2. Некоторые понятия и определения 11 1.3. Линейное программирование 15 Упражнения 21 Литература 22 Глава 2. Алгоритмы построения деревьев 23 2.1. Алгоритмы построения покрывающих деревьев 23 2.2. Алгоритм построения максимального ориентированного леса 30 Упражнения 40 Литература 41 Глава 3. Алгоритмы поиска путей 42 3.1. Алгоритм поиска кратчайшего пути 42 3.2. Алгоритмы поиска всех кратчайших путей 51 3.3. Алгоритм поиска k кратчайших путей 63 3.4. Поиск других оптимальных путей 77 Упражнения 81 Литература 83 Глава 4. Потоковые алгоритмы 84 4.1. Введение 84 4.2. Алгоритм поиска максимального потока 91 4.3. Алгоритм поиска потока минимальной стоимости 100 4.4. Алгоритм дефекта 111 4.5. Алгоритм поиска динамического потока 122 4.6. Потоки с усилениями 146 Упражнения 166 Литература 170 Глава 5. Алгоритмы поиска паросочетаний и покрытий 171 5.1. Введение 171 5.2. Алгоритм решения задачи о паросочетаний максимальной мощности 175 5.3 Алгоритм выбора паросочетания с максимальным весом 189 5.4. Алгоритм построения покрытия с минимальным весом 201 Упражнения 216 Литература 218 Глава 6. Задача почтальона 219 6.1. Введение 219 6.2. Задача почтальона для неориентированного графа 222 6.3. Задача почтальона для ориентированного графа 227
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6.4. Задача почтальона для смешанного графа 231 Упражнения 239 Литература 240 Глава 7. Задача коммивояжера 241 7.1. Формулировка и некоторые свойства решений задачи коммивояжера 241 7.2. Условия существования гамильтонова контура 244 7.3. Нижние границы 251 7.4. Методы решения задачи коммивояжера 255 Упражнения 262 Литература 264 Глава 8. Задачи размещения 265 8.1. Введение 265 8.2. Задачи поиска центра 273 8.3. Задачи поиска медиан 279 8.4. Обобщения 287 Упражнения 288 Литература 289 Глава 9. Сетевые графика 290 9.1. Метод критического пути (МКП) 290 9.2. Определение длительности выполнения операции из условия
обеспечения минимальной стоимости 302
9.3. Обобщенные сетевые графики 309 Упражнения 315 Литература 318 Предметный указатель 319
Предметный указатель Абсолютная медиана 272, 281 Абсолютный центр графа 272, 275 Алгебраическое направление теории
графов 10 Алгоритм 11, 15 - Данцига 51, 57—60, 63, 249 - - обобщенный 74—77 - Дейкстры 44—49, 61—63, 81 - дефекта 111, 113, 117—122, 304 - Джевелла 153 - Флойда 53, 58, 61—63 - - обобщенный 74—77 - Форда 49—51, 61—63 - - и Фалкерсона 92—101 - Эдмондса 178, 189 Анализ вычислительной сложности
60—63
Базисное решение 20, 21 Букет 25 Величина дефекта 116—118 Венгерское дерево 181—183 Вершина 10 - внешняя 180 - внутренняя 180 - конечная 12 - концевая 25 - насыщенная 203, 204 - начальная 12 - ненасыщенная 203, 205 - открытая 179 - паросочетания 179 - пустая 203 Вершинное число 105, 109, 151 Вес дерева 23
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Взвешенное размещение 287 Внутренняя точка 267 Время прохождения 123 Гамильтонов контур 241, 243, 244—
264 - - оптимальный 242, 244—264 - цикл 250 Главная абсолютная медиана 272,
282 - медиана 272, 280 Главный абсолютный центр 272, 275,
278 - центр графа 272, 274, 275 Граф 10 - двудольный 175, 178 - неориентированный 11 - нечетный 222 - связный 13, 14 - сильно связный 244, 246 - четный 221 Дерево 13 - кратчайших путей 45 - минимальной стоимости 23 - ориентированное 254 - чередующееся 180 Динамический поток 123—132 Длина пути, 78 - цепи 12 Дуга 10 - обратная 87, 136 - порождающая спрос 147 - промежуточная 86 - прямая 87, 136 - увеличивающая 85, 89 - уменьшающая 85, 89 Единица потока 84 Задача коммивояжера 241 - - общая 241 - об узких местах 78 - о Кенигсбергских мостах 9 - - максимальном потоке 91, 92, 102 - - паросочетания 11 - - - максимальной мощности 173,
175, 178
- - - минимальной мощности 173 - - - с максимальным весом 172, 175 - - - - минимальным весом 173 - - покрытии максимальной
мощности 172 - - - минимальной мощности 172, 175 - - - с максимальным весом 173 - - - - минимальным весом 173, 175 - - потоке минимальной стоимости
101, 113, 114, 147—149, 151, 152
- - путях с усилениями 79 - поиска медиан 279—285 - - центра 273—279 - почтальона 9, 219—240 - размещения 265—288 Источник 84 Компонент графа 13, 14 - - сильно связный 245 Контур 12, 33 - простой 12, 247 Коэффициент усиления дуги 79, 80,
146 Критическая операция 300 Критический путь 300 Лес 14 - максимальный ориентированный
31, 38, 39 - минимальный 31 Линейное программирование 15—21,
92, 147 - - двойственная задача 18 - - прямая задача 18 Маршрут 219 - коммивояжера 241 - - оптимальный 242, 243 - почтальона 220, 222, 225 Матрица графа 15 - инциденции 15 Медиана 266, 272, 279 Метод ветвей и границ 256—259 - критического пути 290—302 - РЕКТ 301, 302
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- последовательного улучшения решения 256, 260—264
Модель Фалкерсона 302 Неравенство треугольника 242, 243 Обобщенная операция сложения 64 - - сравнения 64 Обратный поиск 67, 73 Окрашивание ребер 24 Оптимальная длина пути 65 Оптимальный поток 155 - путь 77 Оптимизационное направление
теории графов 9 Оценка времени выполнения
операции 301 Паросочетание 171—205 - максимальное по мощности 171,
183 - минимальной мощности 172 - с максимальным весом 172, 189 Петля 12 Подграф 13 - порожденный 13 Поедающий алгоритм 26, 40 Покрывающее дерево 14, 23—29, 31,
32 Покрытие 171, 205—217 Поток лексикографический 144 - наипозднейшего отправления 140,
143 - - прибытия 139 - наискорейшего отправления 140,
141 - - прибытия 132—143 - с усилениями 146, 153 Пропускная способность 84, 91, 95 Прямой поиск 67, 73 Путь 12, 42 - кратчайший 42—59 Разрез 14, 94 - насыщенный 105, 130, 131 - простой 14, 94 Расстояние вершина — вершина 267 - вершина — дуга 268
- точка — вершина 267 - точка — дуга 269 Ребро 11 Резерв времени независимый 299 - - полный 298 - - свободный 298, 299 Решающий узел 310 Свертка вектора 74 Сетевой график 290, 293—315 - - обобщенный 309—315 Сеть 11, 85, 122 - с усилениями 151, 152 Симплекс-алгоритм 21 Степень вершины 220 - - внешняя 221 - - внутренняя 221 - дефектности дуги 116 - захода 21 - исхода 21 Сток 84 Теорема Гуйя-Ури 246 Теория графов 9 Увеличение потока 87 - - максимальное 87, 88, 96 Узкое место 97 Уменьшение потока 87 Уравнение сохранения потока 151,
152 Усиление дуги 146 Условия дополняющей нежесткости
18, 19, 107, 108, 120, 149 - неотрицательности 17 Функция расстояний 281, 282 f-точка 267 Целевая функция 16 Центр графа 266, 272 Цепь 12 - взвешенная увеличивающаяся
чередующаяся 189 - простая 12, 179 - увеличивающаяся 88, 138, 139, 184 - - чередующаяся 179 - чередующаяся 178—180 Цикл 12, 117
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- генерирующий 149, 150, 152 - нечетный 179, 181, 184 - поглощающий 150, 152 - простой 179
Чистый поток 86, 87, 116, 120 Эйлеров маршрут 220, 228, 229, 231,
232
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