how to find all integer solutions to sum(xi) =b, with linear constraints

Question

Suppose sum(xi) = 10, 0<= xi <= 2, i = 1, 2, ..., 10. How to find all integer solutions for xi. thank you. I have read about Euclidean algorithm, but it looks like just for two unknown variables. What algorithms can be used here.

Answer 1

Recursion is best. Here is the natural Python solution with generators:

def solutions(variables, sum_left, max_value):
    if 0 == variables:
        if 0 == sum_left:
            yield []
    else:
        for i in range(0, max_value + 1):
            if sum_left < i:
                break
            else:
                for partial_solution in solutions(variables - 1, sum_left - i,
                                                  max_value):
                    yield [i] + partial_solution

for x in solutions(10, 10, 2):
    print(x)

The benefit of generators being that you don't have to build a long list in memory first. Here is an alternate solution which does not use generators and also avoids building up the list.

def do_something_for_solutions(variables, sum_left, max_value, known=None):
    if known is None:
        known = []
    if 0 == variables:
        if 0 == sum_left:
            do_something(known)
    else:
        for i in range(0, max_value + 1):
            if sum_left < i:
                break
            else:
                do_something_for_solutions(variables - 1, sum_left - i,
                                           max_value, known + [i])

def do_something(solution):
    print(solution)

do_something_for_solutions(10, 10, 2)

If you choose to return solutions, that is possible as follows:

def solutions(variables, sum_left, max_value):
    if 0 == variables:
        if 0 == sum_left:
            return [[]]
        else:
            return []
    else:
        answer = []
        for i in range(0, max_value + 1):
            if sum_left < i:
                break
            else:
                for partial_solution in solutions(variables - 1, sum_left - i,
                                                  max_value):
                    answer.append([i] + partial_solution)
        return answer

for x in solutions(10, 10, 2):
    print(x)

(Be warned that if you change the parameters, that list can easily become huge...)