Some Results on Incremental Vertex Cover Problem

  • Wenqiang Dai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6124)


In the classical k-vertex cover problem, we wish to find a minimum weight set of vertices that covers at least k edges. In the incremental version of the k-vertex cover problem, we wish to find a sequence of vertices, such that if we choose the smallest prefix of vertices in the sequence that covers at least k edges, this solution is close in value to that of the optimal k-vertex cover solution. The maximum ratio is called competitive ratio. Previously the known upper bound of competitive ratio was 4α, where α is the approximation ratio of the k-vertex cover problem. And the known lower bound was 1.36 unless P = NP, or 2 − ε for any constant ε assuming the Unique Game Conjecture. In this paper we present some new results for this problem. Firstly we prove that, without any computational complexity assumption, the lower bound of competitive ratio of incremental vertex cover problem is φ, where \(\phi=\frac{\sqrt{5}+1}{2}\approx 1.618\) is the golden ratio. We then consider the restricted versions where k is restricted to one of two given values(Named 2-IVC problem) and one of three given values(Named 3-IVC problem). For 2-IVC problem, we give an algorithm to prove that the competitive ratio is at most φα. This incremental algorithm is also optimal for 2-IVC problem if we are permitted to use non-polynomial time. For the 3-IVC problem, we give an incremental algorithm with ratio factor \((1+\sqrt{2})\alpha\).


Approximation Algorithm Approximation Ratio Competitive Ratio Cover Problem Vertex Cover 
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  1. 1.
    Bar-Yehuda, R.: Using homogeneous weights for approximating the partial cover problem. Journal of Algorithms 39(2), 137–144 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bar-Yehuda, R., Even, S.: A linear time approximation algorithm for the weighted vertex cover problem. J. of Algorithms 2, 198–203 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–45 (1985)MathSciNetGoogle Scholar
  4. 4.
    Bshouty, N., Burroughs, L.: Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 298–308. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  5. 5.
    Clarkson, K.L.: A modification of the greedy algorithm for the vertex cover. Information Processing Letters 16, 23–25 (1983)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Annuals of Mathematics 162(1) (2005); Preliminary version in STOC 2002 (2002) Google Scholar
  7. 7.
    Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. Journal of Algorithms 53(1), 55–84 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Håstad, J.: Some optimal inapproximability results. J. of ACM 48(4), 798–859 (2001)zbMATHCrossRefGoogle Scholar
  9. 9.
    Hartline, J., Sharp, A.: An Incremental Model for Combinatorial Minimization (2006),
  10. 10.
    Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics 6, 243–254 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hochbaum, D.S. (ed.): Approximation Algorithms for NP-hard Problems. PWS Publishing Company (1997)Google Scholar
  12. 12.
    Hochbaum, D.S.: The t-vertex cover problem: Extending the half integrality framework with budget constraints. In: Jansen, K., Rolim, J.D.P. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 111–122. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  13. 13.
    Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)Google Scholar
  14. 14.
    Khot, S., Regev, O.: Vertex cover might be hard to approximaite to within 2 − ε. In: Proceedings of the 18th IEEE Conference on Computational Complexity (2003)Google Scholar
  15. 15.
    Lin, G.L., Nagarajan, C., Rajamaran, R., Williamson, D.P.: A general approach for incremental approximation and hierarchical clustering. In: Proc. 17th Symp. on Discrete Algorithms (SODA). ACM/SIAM (2006)Google Scholar
  16. 16.
    Mestre, J.: A primal-dual approximation algorithm for partial vertex cover: making educated guesses. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 182–191. Springer, Heidelberg (2005)Google Scholar
  17. 17.
    Nemhauser, G.L., Trotter Jr., L.E.: Vertex packings: Structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Wenqiang Dai
    • 1
  1. 1.School of Management and EconomicsUniversity of Electronic Science and Technology of ChinaChengduP. R.  China

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