On Problems Related to Unbounded SubsetSum: A Unified Combinatorial Approach

Abstract

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 $c={\sum }_{i=1}^n{w}_i{m}_i$. 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.