Journal of Combinatorial Optimization

, Volume 34, Issue 4, pp 1052–1059 | Cite as

Approximation for vertex cover in \(\beta \)-conflict graphs

  • Dongjing Miao
  • Zhipeng CaiEmail author
  • Weitian Tong
  • Jianzhong Li


Conflict graph is a union of finite given sets of disjoint complete multipartite graphs. Vertex cover on this kind of graph is used first to model data inconsistency problems in database application. It is NP-complete if the number of given sets r is fixed, and can be approximated within \(2-\frac{1}{2^r}\) (Miao et al. in Proceedings of the 9th international conference on combinatorial optimization and applications, vol 9486. COCOA 2015, New York. Springer, New York, pp 395–408, 2015). This paper shows a better algorithm to improve the approximation for dense cases. If the ratio of vertex not belongs to any wheel complete multipartite graph is no more than \(\beta <1\), then our algorithm will provide a \((1+\beta +\frac{1-\beta }{k})\)-approximation, where k is a parameter related to degree distribution of wheel complete multipartite graph.


Approximation algorithm Vertex cover Conflict graph Complete multipartite graph 


  1. Bar-Yehuda R, Even S (1985) A local-ratio theorem for approximating the weighted vertex cover problem. Ann Discrete Math 25:27–46MathSciNetzbMATHGoogle Scholar
  2. Cheetham J, Dehne F, Rau-Chaplin A, Stege U, Taillon PJ (2003) Solving large FPT problems on coarse-grained parallel machines. J Comput Syst Sci 67(4):691–706MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chen J, Kanj IA, Xia G (2010) Improved upper bounds for vertex cover. Theor Comput Sci 411(40–42):3736–3756MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dinur I, Safra S (2005) On the hardness of approximating minimum vertex cover. Ann Math 162(1):439–485Google Scholar
  5. Downey RG, Fellows MR (1995) Fixed-parameter tractability and completeness II: on completeness for w[1]. Theor Comput Sci 141(1–2):109–131MathSciNetCrossRefzbMATHGoogle Scholar
  6. Fang J, Huang Z (2010) Reasoning with inconsistent ontologies. Tsinghua Sci Technol 15(6):687–691CrossRefGoogle Scholar
  7. Gavril F (1972) Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J Comput 1(2):180–187MathSciNetCrossRefzbMATHGoogle Scholar
  8. Håstad J (2001) Some optimal inapproximability results. J ACM 48(4):798–859MathSciNetCrossRefzbMATHGoogle Scholar
  9. Karakostas G (2009) A better approximation ratio for the vertex cover problem. ACM Trans Algorithms 5(4):41:1–41:8MathSciNetCrossRefzbMATHGoogle Scholar
  10. Khot S, Regev O (2008) Vertex cover might be hard to approximate to within \(2-\epsilon \). J Comput Syst Sci 74(3):335–349MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kuhn F, Mastrolilli M (2011) Vertex cover in graphs with locally few colors. In: Proceedings of the 38th international colloquim conference on automata, languages and programming, Vol Part I. ICALP’11. Springer, Berlin, pp 498–509Google Scholar
  12. Lin G, Cai Z, Lin D (2006) Vertex covering by paths on trees with its applications in machine translation. Inf Process Lett 97(2):73–81MathSciNetCrossRefzbMATHGoogle Scholar
  13. Liu Y, Wu B, Wang H, Ma P (2014) Bpgm: a big graph mining tool. Tsinghua Sci Technol 19(1):33–38CrossRefGoogle Scholar
  14. Miao D, Li J, Liu X, Gao H (2015) Vertex cover in conflict graphs: complexity and a near optimal approximation. In: Proceedings of the 9th international conference on combinatorial optimization and applications, vol 9486. COCOA 2015, New York. Springer, New York, Inc, pp 395–408Google Scholar
  15. Monien B, Speckenmeyer E (1985) Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Inform 22(1):115–123MathSciNetCrossRefzbMATHGoogle Scholar
  16. Nemhauser GL, Trotter LE Jr (1975) Vertex packings: structural properties and algorithms. Math Program 8(1):232–248MathSciNetCrossRefzbMATHGoogle Scholar
  17. Savage CD (1982) Depth-first search and the vertex cover problem. Inf Process Lett 14(5):233–237MathSciNetCrossRefzbMATHGoogle Scholar
  18. Vazirani VV (2001) Approximation algorithms. Springer, new yorkzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Dongjing Miao
    • 1
  • Zhipeng Cai
    • 1
    Email author
  • Weitian Tong
    • 2
  • Jianzhong Li
    • 3
  1. 1.Department of Computer ScienceGeorgia State UniversityAtlantaUSA
  2. 2.Department of Computer SciencesGeorgia Southern UniversityStatesboroUSA
  3. 3.School of Computer Science and TechnologyHarbin Institute of TechnologyHarbinChina

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