  The Unbounded Subset Sum (USS) problem is an NP-hard computational problem
where the goal is to decide whether there exist non-negative integers $x_1,
\ldots, x_n$ such that $x_1 a_1 + \ldots + x_n a_n = b$, where $a_1 < \cdots <
a_n < b$ are distinct positive integers with $\text{gcd}(a_1, \ldots, a_n)$
dividing $b$. The problem can be solved in pseudopolynomial time, while
specialized cases, such as when $b$ exceeds the Frobenius number of $a_1,
\ldots, a_n$ simplify to a total problem where a solution always exists.
  This paper explores the concept of totality in USS. The challenge in this
setting is to actually find a solution, even though we know its existence is
guaranteed. We focus on the instances of USS where solutions are guaranteed for
large $b$. We show that when $b$ is slightly greater than the Frobenius number,
we can find the solution to USS in polynomial time.
  We then show how our results extend to Integer Programming with Equalities
(ILPE), highlighting conditions under which ILPE becomes total. We investigate
the diagonal Frobenius number, which is the appropriate generalization of the
Frobenius number to this context. In this setting, we give a polynomial-time
algorithm to find a solution of ILPE. The bound obtained from our algorithmic
procedure for finding a solution almost matches the recent existential bound of
Bach, Eisenbrand, Rothvoss, and Weismantel (2024).
