无界SubsetSum相关问题的统一组合方法
On Problems Related to Unbounded SubsetSum: A Unified Combinatorial Approach
摘要：
无界子集求和是一个经典的教科书问题：给定整数$w_1,w_2,\cdots,w_n\in [1,u],~c,u$，我们需要找出是否存在满足$c=\sum_{i=1}^n w_im_i$的$m_1,m_2,\cdots,m_n\in \mathbb{N}$。在它的全目标版本中，给出了$t\in \mathbb{Z}_+$，并且需要所有整数的答案$c\in[0,t]$。在本文中，我们研究了这个简单问题的三种推广：全目标无界背包，全目标硬币兑换和剩余表。通过对解的结构的新的组合见解，我们提出了一种新的两阶段方法来解决这类问题。因此，我们提出了CoinChange和Residue Table的第一个近线性算法，该算法在$\tilde{O}(u+t)$和$\tilde{O}(u)$中时间确定性地运行。我们还表明，如果我们可以计算$(\min,+)$卷积的$n$长度的阵列在$T(n)$时间，那么所有目标无界背包可以解决在$\tilde{O}(T(u)+t)$时间，从而建立次二次等价的所有目标无界背包和$(\min,+)$卷积。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
