subset sum

Given a multiset $S$ of $n$ positive integers and a target integer $t$, the subset sum problem is to decide if there is a subset of $S$ that sums up to $t$. We present a new divide-and-conquer algorithm that computes \emph{all} the realizable subset sums up to an integer $u$ in $\widetilde{O}\!\left(\min\{\sqrt{n}u,u^{4/3},\sigma\}\right)$, where $\sigma$ is the sum of all elements in $S$ and $\widetilde{O}$ hides polylogarithmic factors. This result improves upon the standard dynamic programming algorithm that runs in $O(nu)$ time. To the best of our knowledge, the new algorithm is the fastest general algorithm for this problem. We also present a modified algorithm for cyclic groups, which computes all the realizable subset sums within the group in $\widetilde{O}\!\left(\min\{\sqrt{n}m,m^{5/4}\}\right)$ time, where $m$ is the order of the group.

We introduce a new combinatorial optimization problem in this article, called the minimum common integer partition (MCIP) problem, which was inspired by computational biology applications including ortholog assignment and DNA fingerprint assembly. A partition of a positive integer n is a multiset of positive integers that add up to exactly n , and an integer partition of a multiset S of integers is defined as the multiset union of partitions of integers in S . Given a sequence of multisets S 1 , S 2 , …, S k of integers, where k ≥ 2, we say that a multiset is a common integer partition if it is an integer partition of every multiset S i , 1 ≤ i ≤ k . The MCIP problem is thus defined as to find a common integer partition of S 1 , S 2 , …, S k with the minimum cardinality, denoted as MCIP( S 1 , S 2 , …, S k ). It is easy to see that the MCIP problem is NP-hard, since it generalizes the well-known subset sum problem. We can in fact show that it is APX-hard. We will also present a 5/4-...

Given a multiset S of n positive integers and a target integer t, the subset sum problem is to decide if there is a subset of S that sums up to t. We present a new divide-and-conquer algorithm that computes all the realizable subset sums up to an integer u in O({√(n)u,u^4/3,σ}), where σ is the sum of all elements in S and O hides polylogarithmic factors. This result improves upon the standard dynamic programming algorithm that runs in O(nu) time. To the best of our knowledge, the new algorithm is the fastest general algorithm for this problem. We also present a modified algorithm for cyclic groups, which computes all the realizable subset sums within the group in O({√(n)m,m^5/4}) time, where m is the order of the group.