How to find all combinations that sum up to at most a constant?

Question

Let P=[P1, P2, ..., Pk] be k positive integers and let T be a positive integer. I would like to generate all combinations that sum up

Answer 1

This is like the subset sum problem (mentioned in the comments), except that's about finding numbers summing equal to a target. You want to find numbers summing to a total less-than-or-equal to the target.

Nonetheless, a similar approach with dynamic programming can be used:

def subset_sum(vals, target=0):
    sums = {0: [()]}  # key=sum, value=list of subsets for the sum
    if target in sums:
        yield from sums[target]  # annoying base case
    for val in vals:
        items = sums.items()  # don't change dict size during iteration
        sums = dict(items)
        for prev_sum, prev_subsets in items:
            sum_ = prev_sum + val
            subsets = [s + (val,) for s in prev_subsets]
            sums[sum_] = sums.get(sum_, []) + subsets
            if sum_ <= target:
                yield from subsets

Demo:

>>> for subset in subset_sum([1, 2, 3], target=4):
...     print(*subset, sep='+')
...     
1
2
1+2
3
1+3