On the minimum common integer partition problem

Abstract. We introduce a new combinatorial optimization problem in this paper, called theMinimum Common Integer Partition (MCIP) prob-lem, which was inspired by computational biology applications including ortholog assignment and DNA fingerprint assembly. A partition of a pos-itive 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 mul-tisets 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 5 4-approximation algorithm for the MCIP problem when k = 2, and a 3k(k−1) 3k−2-approximation algorithm for k ≥ 3.