Fast Knapsack Solver适用于大问题

Question

我想用Python大致解决大数据集的背包问题。

现在，我正在使用this implementation，这适用于以下小例子：

import knapsack
weight = np.random.randint(10, size = 10)
value = np.random.randint(10, size = 10)
capacity = 5
knapsack.knapsack(weight, value).solve(capacity)

但是当我们将它扩展到：

import knapsack
weight = np.random.randint(10, size = 1000)
value = np.random.randint(10, size = 1000)
capacity = 500
knapsack.knapsack(weight, value).solve(capacity)

程序刚刚卡住并发出错误。我想知道是否有一些背包问题的实现，我们可以在10秒钟内说出计算内容并返回到目前为止找到的最佳解决方案，这可能吗？

Answer 1

这里有0-1背包的小型原型0-1整数编程方法！

此代码：

• 并不是完美无瑕！
  • e.g。约束与边界（后者更有效;但是懒得再次检查cylp;过去的问题）
• 对窗户的支持不多！
  • windows用户：转到pulp，它会带来相同的求解器（imho是最好的免费开源MIP求解器）;虽然造型看起来很不一样！
• 没有调整！
  • 观察：在求解器Cgl中使用的CoinOR Cbc支持额外的背包切割！
  • 如日志所示：示例过于简单，无法使用它们！
• 有界/无界背包版本只需修改界限即可轻松处理

这里的例子只解决了OP使用1的PRNG种子定义的一个问题，其中 0.02秒，但这不是科学测试！ NP难问题都是关于简单与硬实例（巨大的差异！），因此，要检查的数据非常重要！可以观察到，这个例子没有真正的完整性差距。

代码

import numpy as np
import scipy.sparse as sp
from cylp.cy import CyClpSimplex
np.random.seed(1)

""" INSTANCE """
weight = np.random.randint(10, size = 1000)
value = np.random.randint(10, size = 1000)
capacity = 500

""" SOLVE """
n = weight.shape[0]
model = CyClpSimplex()
x = model.addVariable('x', n, isInt=True)
model.objective = -value
model += sp.eye(n) * x >= np.zeros(n)  # could be improved
model += sp.eye(n) * x <= np.ones(n)   # """
model += np.matrix(weight) * x <= capacity  # cylp somewhat outdated in terms of np-usage!
cbcModel = model.getCbcModel()  # Clp -> Cbc model / LP -> MIP
cbcModel.logLevel = True
status = cbcModel.solve()
x_sol = np.array(cbcModel.primalVariableSolution['x'].round()).astype(int)  # assumes there is one

print(x_sol)
print(x_sol.dot(weight))
print(x_sol.dot(value))

输出

Welcome to the CBC MILP Solver 
Version: 2.9.9 
Build Date: Jan 15 2018 

command line - ICbcModel -solve -quit (default strategy 1)
Continuous objective value is -1965.33 - 0.00 seconds
Cgl0004I processed model has 1 rows, 542 columns (542 integer (366 of which binary)) and 542 elements
Cutoff increment increased from 1e-05 to 0.9999
Cbc0038I Initial state - 1 integers unsatisfied sum - 0.333333
Cbc0038I Pass   1: suminf.    0.25000 (1) obj. -1965 iterations 1
Cbc0038I Solution found of -1965
Cbc0038I Branch and bound needed to clear up 1 general integers
Cbc0038I Full problem 1 rows 542 columns, reduced to 1 rows 128 columns
Cbc0038I Cleaned solution of -1965
Cbc0038I Before mini branch and bound, 540 integers at bound fixed and 0 continuous
Cbc0038I Mini branch and bound did not improve solution (0.02 seconds)
Cbc0038I After 0.02 seconds - Feasibility pump exiting with objective of -1965 - took 0.01 seconds
Cbc0012I Integer solution of -1965 found by feasibility pump after 0 iterations and 0 nodes (0.02 seconds)
Cbc0038I Full problem 1 rows 542 columns, reduced to 1 rows 2 columns
Cbc0001I Search completed - best objective -1965, took 0 iterations and 0 nodes (0.02 seconds)
Cbc0035I Maximum depth 0, 362 variables fixed on reduced cost
Cuts at root node changed objective from -1965.33 to -1965.33
Probing was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
Gomory was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
Knapsack was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
Clique was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
MixedIntegerRounding2 was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
FlowCover was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)
TwoMirCuts was tried 0 times and created 0 cuts of which 0 were active after adding rounds of cuts (0.000 seconds)

Result - Optimal solution found

Objective value:                -1965.00000000
Enumerated nodes:               0
Total iterations:               0
Time (CPU seconds):             0.02
Time (Wallclock seconds):       0.02

Total time (CPU seconds):       0.02   (Wallclock seconds):       0.02

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