  Unbounded SubsetSum is a classical textbook problem: given integers
$w_1,w_2,\cdots,w_n\in [1,u],~c,u$, we need to find if there exists
$m_1,m_2,\cdots,m_n\in \mathbb{N}$ satisfying $c=\sum_{i=1}^n w_im_i$. In its
all-target version, $t\in \mathbb{Z}_+$ is given and answer for all integers
$c\in[0,t]$ is required. In this paper, we study three generalizations of this
simple problem: All-Target Unbounded Knapsack, All-Target CoinChange and
Residue Table. By new combinatorial insights into the structures of solutions,
we present a novel two-phase approach for such problems. As a result, we
present the first near-linear algorithms for CoinChange and Residue Table,
which runs in $\tilde{O}(u+t)$ and $\tilde{O}(u)$ time deterministically. We
also show if we can compute $(\min,+)$ convolution for $n$-length arrays in
$T(n)$ time, then All-Target Unbounded Knapsack can be solved in
$\tilde{O}(T(u)+t)$ time, thus establishing sub-quadratic equivalence between
All-Target Unbounded Knapsack and $(\min,+)$ convolution.
