An Improved Pseudopolynomial Time Algorithm for Subset Sum

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.