考虑到这些测试用例：
votes = [6]*28
m = 10
votes1 = [5]*28+[6]*2
m1 = 10
votes2 = [5]*29+[10]*1
m2 = 10
votes3 = [8, 8, 16, 12, 12, 12, 4, 4, 12, 4, 4, 4, 8, 12, 12, 8, 8, 16, 12, 4, 16, 16, 12, 16, 12, 16, 12, 4, 16, 4, 4, 12, 4, 12, 12, 4, 16, 12, 16, 8]
m3 = 20
votes4 = [22, 21, 34, 39, 28, 33, 32, 40, 22, 34, 36, 27, 37, 34, 40, 38, 39, 32, 37, 40, 31, 37, 22, 21, 35, 34, 24, 40, 34, 21, 24, 20, 30, 31, 22, 30, 31, 25, 20, 38, 24, 23, 32, 27, 20, 31, 27, 32, 22, 32, 33, 34, 40, 38, 36, 29, 34, 24, 24, 39, 32, 37, 30, 20, 29, 26, 36, 40, 34, 22, 30, 27, 38, 27, 26, 28, 23, 40, 31, 22, 23, 35, 23, 31, 23, 39, 30, 20, 20, 35, 27, 23, 23, 29, 40, 20, 34, 40, 28, 25]
m4 = 50
votes5 = [25, 25, 25, 24, 25, 24, 24, 25, 26, 25, 26, 24, 25, 26, 24, 26, 24, 26, 26, 25, 26, 24, 26, 24, 26, 26, 26, 25, 25, 26, 24, 26, 25, 25, 24, 25, 25, 26, 26, 26, 25, 26, 25, 26, 25, 25, 24, 24, 24, 25, 24, 26, 25, 24, 26, 24, 24, 26, 24, 26, 24, 24, 24, 26, 24, 25, 24, 26, 25, 25, 26, 25, 25, 25, 25, 26, 25, 24, 25, 25, 24, 24, 24, 26, 26, 26, 25, 24, 25, 25, 25, 26, 25, 24, 26, 24, 25, 26, 24, 26]
m5 = 50
给出以下界限：
def upperbound(v, m, n):
return math.floor(v - 0.25 * m * n - 1)
def lowerbound(m, n):
return math.ceil(0.25 * m * n + 1)
我想找出是否有一个长度恰好为len(votes)/2的子集，它将满足给定的upperbound和lowerbound。
以下是我使用背包解决问题的尝试，但没有考虑子集的长度。
import math
def winnable(votes, m):
n = len(votes) # Number of columns
v = sum(votes)
ub = upperbound(v, m, n)
lb = lowerbound(m, n)
max_possible = knapSack(ub, votes, n)
if max_possible < lb:
return "not possible"
else:
return "possible"
def knapSack(ub, val, n):
K = [[0 for x in range(ub + 1)] for x in range(n + 1)]
# Build table K[][] in bottom up manner
for i in range(n + 1):
for w in range(ub + 1):
if i == 0 or w == 0:
K[i][w] = 0
elif val[i - 1] <= w:
K[i][w] = max(val[i - 1] + K[i - 1][w - val[i - 1]], K[i - 1][w])
else:
K[i][w] = K[i - 1][w]
return K[n][ub]
是否可以进一步修改我的解决方案，以考虑子集中元素的数量。
答案 0 :(得分：2)
我已经实施了@mrmcgreg关于为问题添加额外维度的建议
import math
def has_valid_subset(votes, m):
n = len(votes)
sum_min = math.ceil(0.25 * m * n + 1)
sum_max = math.floor(sum(votes) - 0.25 * m * n - 1)
n_half = n // 2
K = [[[(False, 0) for elements in range(min(n_half + 1, index + 1))]
for index in range(len(votes) + 1)]
for weight in range(sum_max + 1)]
for weight in range(sum_max + 1):
for index in range(len(votes) + 1):
if index == 0:
K[weight][index][0] = (True, 0)
continue
v = votes[index - 1]
for elements in range(min(n_half + 1, index)):
if v > weight:
K[weight][index][elements] = K[weight][index - 1][elements]
else:
skip_ok, skip_w = K[weight][index - 1][elements]
add_ok, add_prev_w = K[weight - v][index - 1][elements - 1]
add_w = add_prev_w + v
if skip_ok and add_ok:
K[weight][index][elements] = (True, max(skip_w, add_w))
elif skip_ok:
K[weight][index][elements] = (True, skip_w)
elif add_ok:
K[weight][index][elements] = (True, add_w)
b_max, w_max = K[-1][-1][-1]
if not b_max:
return False
return w_max >= sum_min
试验：
votes = [6]*28
m = 10
print(has_valid_subset(votes, m))
# True
votes1 = [5]*28+[6]*2
m1 = 10
print(has_valid_subset(votes1, m1))
# True
votes2 = [5]*29+[10]*1
m2 = 10
print(has_valid_subset(votes2, m2))
# False
votes3 = [8, 8, 16, 12, 12, 12, 4, 4, 12, 4, 4, 4, 8, 12, 12, 8, 8, 16, 12, 4, 16, 16, 12, 16, 12, 16, 12, 4, 16, 4, 4, 12, 4, 12, 12, 4, 16, 12, 16, 8]
m3 = 20
print(has_valid_subset(votes3, m3))
# False
votes4 = [22, 21, 34, 39, 28, 33, 32, 40, 22, 34, 36, 27, 37, 34, 40, 38, 39, 32, 37, 40, 31, 37, 22, 21, 35, 34, 24, 40, 34, 21, 24, 20, 30, 31, 22, 30, 31, 25, 20, 38, 24, 23, 32, 27, 20, 31, 27, 32, 22, 32, 33, 34, 40, 38, 36, 29, 34, 24, 24, 39, 32, 37, 30, 20, 29, 26, 36, 40, 34, 22, 30, 27, 38, 27, 26, 28, 23, 40, 31, 22, 23, 35, 23, 31, 23, 39, 30, 20, 20, 35, 27, 23, 23, 29, 40, 20, 34, 40, 28, 25]
m4 = 50
print(has_valid_subset(votes4, m4))
# True
votes5 = [25, 25, 25, 24, 25, 24, 24, 25, 26, 25, 26, 24, 25, 26, 24, 26, 24, 26, 26, 25, 26, 24, 26, 24, 26, 26, 26, 25, 25, 26, 24, 26, 25, 25, 24, 25, 25, 26, 26, 26, 25, 26, 25, 26, 25, 25, 24, 24, 24, 25, 24, 26, 25, 24, 26, 24, 24, 26, 24, 26, 24, 24, 24, 26, 24, 25, 24, 26, 25, 25, 26, 25, 25, 25, 25, 26, 25, 24, 25, 25, 24, 24, 24, 26, 26, 26, 25, 24, 25, 25, 25, 26, 25, 24, 26, 24, 25, 26, 24, 26]
m5 = 50
print(has_valid_subset(votes5, m5))
# True
答案 1 :(得分：1)
这是一个合理有效的解决方案的核心。经过轻微测试但可能是正确的。 （最大的问题，您是否希望解决方案包含边界或排除它们？）
def subset_sum_of_len_in_range (vals, total_to_use, lower, upper):
# Sorting it makes it easy to calculate min/max of partial sum
# from here - it will be the beginning or end. This will be
# useful in filtering.
sorted_vals = sorted(vals)
# Precalculating partial sums from beginning makes partial sum
# of a range even easier - just subtract two.
total = 0
cum_prev_sum = [total]
for i in sorted_vals:
total = total + i
cum_prev_sum.append(total)
# It is always easiest to solve DP problems by caching recursive ones.
cache = {}
# And now our recursive cached solver.
def sub_problem (position, to_use, current_sum):
if len(vals) - position < to_use:
# Not enough values left to possibly solve this.
return False
cache_key = (position, to_use, current_sum)
if cache_key not in cache:
lowest_sum = current_sum + cum_prev_sum[position + to_use] - cum_prev_sum[position]
if upper < lowest_sum:
# Can't possibly get in range.
cache[cache_key] = False
return False
elif lower <= lowest_sum:
# Found one in range!
cache[cache_key] = True
return True
highest_sum = current_sum + cum_prev_sum[len(vals)] - cum_prev_sum[len(vals) - to_use]
if highest_sum < lower:
# Can't possibly get in range.
cache[cache_key] = False
return False
elif highest_sum <= upper:
# Found one in range!
cache[cache_key] = True
return True
# Now try recursion.
if sub_problem(position + 1, to_use, current_sum):
# There is a solution that did not use this value
cache[cache_key] = True
elif sub_problem(position + 1, to_use-1, current_sum + vals[position]):
# There is a solution that did use this value
cache[cache_key] = True
else:
# There is no solution.
cache[cache_key] = False
return cache[cache_key]
return sub_problem(0, total_to_use, 0)
将它用于解决您的问题......
import math
def upperbound(v, m, n):
return math.floor(v - 0.25 * m * n - 1)
def lowerbound(m, n):
return math.ceil(0.25 * m * n + 1)
def winnable(votes, m):
n = len(votes) # Number of columns
v = sum(votes)
ub = upperbound(v, m, n)
lb = lowerbound(m, n)
if subset_sum_of_len_in_range(votes, n/2, lb, ub):
return "possible"
else:
return "not possible"