On the Minimum Common Integer Partition Problem

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


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.


Approximation Algorithm Combinatorial Optimization Problem Partition Problem Minimum Cardinality Input String 


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© Springer-Verlag Berlin Heidelberg 2006

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

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