On the Minimum Common Integer Partition Problem

We introduce a new combinatorial optimization problem in this paper, 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 S1, ⋯, Sk of integers, where k ≥ 2, we say that a multiset is a common integer partition if it is an integer partition of every multiset Si, 1≤ i≤ k. The MCIP problem is thus defined as to find a common integer partition of S1, ⋯, Sk with the minimum cardinality. It is easy to see that the MCIP problem is NP-hard since it generalizes the well-known Set Partition problem. We can in fact show that it is APX-hard. We will also present a $\frac{5}{4}$-approximation algorithm for the MCIP problem when k = 2, and a $\frac{3k(k-1)}{3k-2}$-approximation algorithm for k ≥ 3.