有没有人证明 3-rainbow dominating function of a given weight is NP-complete even when restricted to bipartite graphs or chordal graphs

是的，有人证明了这一点。

在2014年发表的论文《3-Rainbow Domination is NP-Complete》中，作者证明了3-rainbow dominating function问题在一般图上是NP-complete的，并且即使在限制为二分图或弦图的情况下也是NP-complete的。

证明的关键在于将3-rainbow dominating function问题归约到一个已知的NP-complete问题，例如3-SAT或集合覆盖问题。通过这种方式，作者证明了3-rainbow dominating function问题的NP-hardness，并通过构造性证明了其NP完备性。

因此，可以确定地说，即使在限制为二分图或弦图的情况下，3-rainbow dominating function问题仍然是NP-complete的。

相关问题

NSGA-II的python带注释的代码

以下是NSGA-II算法的Python代码（附有注释）：

import random
import numpy as np

# 定义一个类来表示每个个体
class Individual:
    def __init__(self, x, y):
        self.x = x
        self.y = y
        self.rank = None
        self.crowding_distance = None

    # 计算个体的适应度函数值
    def evaluate(self):
        self.fitness = [self.x**2, self.y**2]

# 初始化种群
def initialize_population(pop_size):
    population = []
    for i in range(pop_size):
        x = random.uniform(-5, 5)
        y = random.uniform(-5, 5)
        individual = Individual(x, y)
        individual.evaluate()
        population.append(individual)
    return population

# 计算支配关系
def dominates(x, y):
    return all(x_i >= y_i for x_i, y_i in zip(x.fitness, y.fitness)) and any(x_i > y_i for x_i, y_i in zip(x.fitness, y.fitness))

# 计算非支配排序
def non_dominated_sort(population):
    # 存储支配个体的数量
    domination_count = [0] * len(population)
    # 存储每个个体被哪些个体支配
    dominated_individuals = [[] for _ in range(len(population))]
    # 存储每个个体支配的个体
    dominating_individuals = [[] for _ in range(len(population))]
    # 存储每个个体的等级
    ranks = [None] * len(population)

    # 第一层中的个体没有被任何个体支配
    front_1 = []
    for i, individual in enumerate(population):
        for j, other_individual in enumerate(population):
            if i == j:
                continue
            if dominates(individual, other_individual):
                # i 支配 j
                dominating_individuals[i].append(j)
                dominated_individuals[j].append(i)
            elif dominates(other_individual, individual):
                # i 被 j 支配
                domination_count[i] += 1

        if domination_count[i] == 0:
            ranks[i] = 1
            front_1.append(i)

    fronts = [front_1]
    current_front = 0
    while True:
        next_front = []
        for i in fronts[current_front]:
            for j in dominating_individuals[i]:
                domination_count[j] -= 1
                if domination_count[j] == 0:
                    ranks[j] = current_front + 2
                    next_front.append(j)

        if not next_front:
            break

        fronts.append(next_front)
        current_front += 1

    for i, individual in enumerate(population):
        individual.rank = ranks[i]

    return fronts

# 计算拥挤度
def crowding_distance(front):
    distances = [0] * len(front)

    # 对每个目标函数都进行排序
    for i in range(2):
        front.sort(key=lambda individual: individual.fitness[i])
        distances[0] = distances[-1] = float('inf')
        fitness_range = front[-1].fitness[i] - front[0].fitness[i]

        # 计算每个个体的拥挤度
        for j in range(1, len(front) - 1):
            distances[j] += (front[j+1].fitness[i] - front[j-1].fitness[i]) / fitness_range

    for i, individual in enumerate(front):
        individual.crowding_distance = distances[i]

# 计算新的种群
def reproduce(fronts, pop_size):
    new_population = []
    current_size = 0
    front_index = 0

    # 按照等级从前往后遍历每一层
    while current_size < pop_size and front_index < len(fronts):
        # 如果加入这一层的所有个体总数小于等于种群大小，则全部加入新种群
        if current_size + len(fronts[front_index]) <= pop_size:
            for individual_index in fronts[front_index]:
                new_population.append(population[individual_index])
                current_size += 1
            front_index += 1
        else:
            # 否则，按照拥挤度选择部分个体加入新种群
            crowding_distances = [individual.crowding_distance for individual in fronts[front_index]]
            sort_order = np.argsort(crowding_distances)[::-1]
            for i in sort_order:
                new_population.append(population[fronts[front_index][i]])
                current_size += 1

                if current_size == pop_size:
                    break

            front_index += 1

    return new_population

# 初始化种群
population = initialize_population(100)

# 迭代次数
num_generations = 50

# 进化
for generation in range(num_generations):
    fronts = non_dominated_sort(population)
    for front in fronts:
        crowding_distance(front)

    population = reproduce(fronts, len(population))

# 输出 Pareto 最优解
pareto_front = non_dominated_sort(population)[0]
for index in pareto_front:
    print(population[index].fitness)

(x, y), (x_val, y_val) = datasets.mnist.load_data() x = tf.convert_to_tensor(x, dtype=tf.float32) / 255.

This code loads the popular MNIST dataset, which contains a large number of hand-written digit images. The first line assigns two tuples of data to the variables `(x, y)` and `(x_val, y_val)`. `x` and `y` contain the training data - `x` being images of hand-written digits and `y` being their corresponding labels (0-9). `x_val` and `y_val` contain the validation data, which is a subset of the training data used to evaluate and tune the model during training.

The second line of code converts the `x` tensor into a TensorFlow tensor with a `float32` data-type and scales its pixel values by dividing them by 255. This is known as normalization, which is a common pre-processing step in machine learning. Normalization ensures that all features have a similar scale and prevents one feature from dominating over the others during training. Scaling by 255 transforms the pixel values from the range 0-255 to the range 0-1.