We investigate pseudo-polynomial time algorithms for Subset Sum. Given a multi-set $X$ of $n$ positive integers and a target $t$, Subset Sum asks whether some subset of $X$ sums to $t$. Bringmann proposes an $\tilde{O}(n + t)$-time algorithm [Bringmann SODA'17], and an open question has naturally arisen: can Subset Sum be solved in $O(n + w)$ time? Here $w$ is the maximum integer in $X$. We make a progress towards resolving the open question by proposing an $\tilde{O}(n + \sqrt{wt})$-time algorithm.

翻译：本文研究子集和问题的伪多项式时间算法。给定一个包含$n$个正整数的多重集$X$和目标值$t$，子集和问题询问是否存在$X$的某个子集其元素之和等于$t$。Bringmann提出了一个时间复杂度为$\tilde{O}(n + t)$的算法[Bringmann SODA'17]，由此自然产生了一个开放问题：能否在$O(n + w)$时间内解决子集和问题？其中$w$是$X$中最大整数。我们通过提出一个$\tilde{O}(n + \sqrt{wt})$时间复杂度的算法，在解决该开放问题上取得进展。