# Solution: Subset Sum Using Branch-and-Bound

## Problem

“Given a vector of ints or floats V, find the number of length 3 subsets which have a sum < X. Assume that V contains no duplicates.”

Example: V = [1,2,3,4,5] and X = 10

Valid subsets: [1,2,3], [1,2,4], [1,2,5], [1,3,4], [1,3,5], [2,3,4]

## Solution

We can easily solve this problem for any subset length N using branch-and-bound. Branch-and-bound can be either implemented iteratively (e.g., using depth-first search and a stack) or recursively. I opted for the recursive approach as it is more concise and easier to reason about (in my opinion, at least).

A sample solution in Python is presented below. I have also created a Gist for convenience.

Note that the problem only requires that you find the number of valid subsets, but you can easily extend the solution below to return all valid subsets instead of the count. I leave this as an exercise for the reader.

``````"""
Problem:

Given a vector of ints/floats V, find the number of
length 3 subsets which have a sum < X.

Assume that V contains no duplicates.
"""
def num_subsets(V, X, N):
global count
count = 0

def branch_bound(V, X, N, i=0, s=0, t=0):
"""
Recursive branch-and-bound algorithm for subset sum.

V, X, N: inputs based on the problem (N is subset size)
i: current item index in V
s: current sum
t: number of items taken from V

Result is located in count variable defined above.
"""
# Number of items taken is N -> valid subset found
# Update overall count and return
if t == N:
global count
count += 1
return

# Depth limit for search; abort
elif i == len(V):
return

else:
# Take the current item *only* if it meets the constraint
if s + V[i] < X:
branch_bound(V, X, N, i+1, s+V[i], t+1)

# Always try *not taking* the current item
branch_bound(V, X, N, i+1, s, t)

# Run branch-and-bound
branch_bound(V, X, N)

# Return the number of valid subsets found
return count

if __name__ == '__main__':
# Sample with answer = 6
# Subsets: [1,2,3], [1,2,4], [1,2,5], [1,3,4], [1,3,5], [2,3,4]
V = [1, 2, 3, 4, 5]
X = 10
print('Valid subsets: {0}'.format(num_subsets(V, X, 3)))

# Larger problem with negative numbers and unsorted
# Max subset size is 10
V = [-10, 5, 4, 52, 77, 134, 22, 100, -59, 2, 16, 81, 3, 8, 21]
X = 150
print('Valid subsets: {0}'.format(num_subsets(V, X, 10)))
``````