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 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Heidelberg (1999)MATHGoogle Scholar
  2. 2.
    Andrews, G.E.: The Theory of Partitions. Addison-Wesley, Reading (1976)MATHGoogle Scholar
  3. 3.
    Arkin, E.M., Hassin, R.: On local search for weighted packing problems. Math. Oper. Res. 23, 640–648 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Andrews, G.E., Eriksson, K.: The Integer Partitions. Cambridge (2004)Google Scholar
  5. 5.
    Altschul, S., Lipman, D.: Trees, stars, and multiple sequence alignment. SIAM Journal on Applied Math. 49(1), 197–209 (1989)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chrobak, M., Lolman, P., Sgall, J.: The greedy algorithm for the minimum common string partition problem. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 84–95. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithms, 2nd edn., p. 1017. The MIT Press, Cambridge (2001)MATHGoogle Scholar
  8. 8.
    Chen, X.: The minimum common partition problem revisited (manuscript, 2005)Google Scholar
  9. 9.
    Chen, X., Zheng, J., Fu, Z., Nan, P., Zhong, Y., Lonardi, S., Jiang, T.: Computing the assignment of orthologous genes via genome rearrangement. In: Proc. of 3rd Asia Pacific Bioinformatics Conference (APBC 2005), pp. 363–378 (2005)Google Scholar
  10. 10.
    Chen, X., Zheng, J., Fu, Z., Nan, P., Zhong, Y., Lonardi, S., Jiang, T.: The assignment of orthologous genes via genome rearrangement. IEEE/ACM Transactions on Computational Biology and Bioinformatics 2(4), 302–315 (2005)CrossRefGoogle Scholar
  11. 11.
    Fu, Z.: Assignment of orthologous genes for multichromosomal genomes using genome rearrangement. UCR CS Technical report (2004)Google Scholar
  12. 12.
    Gusfield, D.: Algorithms on Strings, Tree, and Sequences: Computer Science and Computational Biology. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  13. 13.
    Goldstein, A., Kolman, P., Zheng, J.: Minimum common string partition problem: hardness and approximations. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 484–495. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Hannenhalli, S., Pevzner, P.A.: Transforming cabbage into turnip (polynomial algorithm for sorting signed permutations by reversals). In: Proc. 27th Ann. ACM Symp. Theory of Comput. (STOC 1995), pp. 178–189 (1995)Google Scholar
  15. 15.
    Hurkens, C., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM J. Discrete Mathematics 2, 68–72 (1989)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kolman, P.: Approximating reversal distance for strings with bounded number of duplicates in linear time. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 580–590. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Kann, V.: Maximum bounded 3-dimensional matching is MAX SNP-complete. Information Processing Letters 37, 27–35 (1991)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Computer and System Sciences 43, 425–440 (1991)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Remm, M., Storm, C., Sonnhammer, E.: Automatic clustering of orthologs and in-paralogs from pairwise species comparisons. J. Mol. Biol. 314, 1041–1052 (2001)CrossRefGoogle Scholar
  20. 20.
    Sankoff, D.: Mechanisms of genome evolution: models and inference. Bull. Int. Stat. Instit. 47, 461–475 (1989)MathSciNetGoogle Scholar
  21. 21.
    Valinsky, L., Scupham, A., Vedova, G.D., Liu, Z., Figueroa, A., Jampachaisri, K., Yin, B., Bent, E., Mancini-Jones, R., Press, J., Jiang, T., Borneman, J.: Oligonucleotide Fingerprinting of Ribosomal RNA Genes (OFRG). In: Kowalchuk, G.A., de Bruijn, F.J., Head, I.M., Akkermans, A.D.L., van Elsas, J.D. (eds.) Molecular Microbial Ecology Manual, 2nd edn., pp. 569–585. Kluwer Academic Publishers, Dordrecht (2004)Google Scholar
  22. 22.

Copyright information

© 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

Personalised recommendations