  Benchmark instances for the unbounded knapsack problem are typically
generated according to specific criteria within a given constant range $R$, and
these instances can be referred to as the unbounded knapsack problem with
bounded coefficients (UKPB). In order to increase the difficulty of solving
these instances, the knapsack capacity $C$ is usually set to a very large
value. Therefore, an exact algorithm that neither time complexity nor space
complexity includes the capacity coefficient $C$ is highly anticipated.
  In this paper, we propose an exact algorithm with time complexity of $O(R^4)$
and space complexity of $O(R^3)$. The algorithm initially divides the multiset
$N$ into two multisubsets, $N_1$ and $N_2$, based on the profit density of
their types. For the multisubset $N_2$ composed of types with profit density
lower than the maximum profit density type, we utilize a recent branch and
bound (B\&B) result by Dey et al. (Math. Prog., pp 569-587, 2023) to determine
the maximum selection number for types in $N_2$. We then employ the
Unbounded-DP algorithm to exactly solve for the types in $N_2$. For the
multisubset $N_1$ composed of the maximum profit density type and its
counterparts with the same profit density, we transform it into a linear
Diophantine equation and leverage relevant conclusions from the Frobenius
problem to solve it efficiently. In particular, the proof techniques required
by the algorithm are primarily covered in the first-year mathematics
curriculum, which is convenient for subsequent researchers to grasp.
