The Minimum Generalized Vertex Cover Problem

  • Refael Hassin
  • Asaf Levin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2832)

Abstract

Let G=(V,E) be an undirected graph, with three numbers d 0(e) ≥ d 1(e) ≥ d 2(e) ≥ 0 for each edge e ∈ E. A solution is a subset U ⊆ V and d i (e) represents the cost contributed to the solution by the edge e if exactly i of its endpoints are in the solution. The cost of including a vertex v in the solution is c(v). A solution has cost that is equal to the sum of the vertex costs and the edge costs. The minimum generalized vertex cover problem is to compute a minimum cost set of vertices. We study the complexity of the problem when the costs d 0(e)=1, d 1(e)=α and d 2(e)=0 ∀  e  ∈ E and c(v) = β ∀  v  ∈ V for all possible values of α and β. We also provide a pair of 2-approximation algorithms for the general case.

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References

  1. 1.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)MATHGoogle Scholar
  2. 2.
    Hochbaum, D.S.: Solving integer programs over monotone inequalities in three variables: A framework for half integrality and good approximations. European Journal of Operational Research 140, 291–321 (2002)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Krumke, S.O., Marathe, M.V., Noltemeier, H., Ravi, R., Ravi, S.S., Sundaram, R., Wirth, H.C.: Improving minimum cost spanning trees by upgrading nodes. Journal of Algorithms 33, 92–111 (1999)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston (1976)Google Scholar
  5. 5.
    Nemhauser, G.L., Trotter Jr., L.E.: Vertex packing: structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Paik, D., Sahni, S.: Network upgrading problems. Networks 26, 45–58 (1995)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Yannakakis, M.: Edge deletion problems. SIAM J. Computing 10, 297–309 (1981)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Refael Hassin
    • 1
  • Asaf Levin
    • 1
  1. 1.Department of Statistics and Operations ResearchTel-Aviv UniversityTel-AvivIsrael

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