CIAC 2006: Algorithms and Complexity pp 236-247

# On the Minimum Common Integer Partition Problem

• Xin Chen
• Lan Liu
• Zheng Liu
• Tao Jiang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3998)

## Abstract

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 S 1, ⋯, 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≤ ik. The MCIP problem is thus defined as to find a common integer partition of S 1, ⋯, S k 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.

## Keywords

Approximation Algorithm Combinatorial Optimization Problem Partition Problem Minimum Cardinality Input String
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Xin Chen
• 1
• Lan Liu
• 2
• Zheng Liu
• 2
• Tao Jiang
• 2
• 3
1. 1.School of Physical and Mathematical SciencesNanyang Tech. Univ.Singapore
2. 2.Department of Computer ScienceUniv. of California at RiversideUSA
3. 3.Currently visiting at Tsinghua UniversityBeijingChina