Online Maximum k-Coverage

  • Giorgio Ausiello
  • Nicolas Boria
  • Aristotelis Giannakos
  • Giorgio Lucarelli
  • Vangelis Th. Paschos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6914)

Abstract

We study an online model for the maximum k-vertex-coverage problem, where given a graph G = (V,E) and an integer k, we ask for a subset A ⊆ V, such that |A| = k and the number of edges covered by A is maximized. In our model, at each step i, a new vertex vi is revealed, and we have to decide whether we will keep it or discard it. At any time of the process, only k vertices can be kept in memory; if at some point the current solution already contains k vertices, any inclusion of a new vertex in the solution must entail the definite deletion of another vertex of the current solution (a vertex not kept when revealed is definitely deleted). We propose algorithms for several natural classes of graphs (mainly regular and bipartite), improving on an easy \(\frac{1}{2}\)-competitive ratio. We next settle a set-version of the problem, called maximum k-(set)-coverage problem. For this problem we present an algorithm that improves upon former results for the same model for small and moderate values of k.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Saha, B., Getoor, L.: On maximum coverage in the streaming model & application to multi-topic blog-watch. In: DM 2009, pp. 697–708. SIAM, Philadelphia (2009)Google Scholar
  2. 2.
    Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45, 634–652 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Hochbaum, D.S., Pathria, A.: Analysis of the greedy approach in problems of maximum k-coverage. Naval Research Logistics 45, 615–627 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ageev, A.A., Sviridenko, M.I.: Approximation algorithms for maximum coverage and max cut with given sizes of parts. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds.) IPCO 1999. LNCS, vol. 1610, pp. 17–30. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Feige, U., Langberg, M.: Approximation algorithms for maximization problems arising in graph partitioning. Journal of Algorithms 41, 174–211 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Han, Q., Ye, Y., Zhang, H., Zhang, J.: On approximation of max-vertex-cover. European Journal of Operational Research 143, 342–355 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Corneil, D.G., Perl, Y.: Clustering and domination in perfect graphs. Discrete Applied Mathematics 9, 27–39 (1984)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Rader Jr., D.J., Woeginger, G.J.: The quadratic 0-1 knapsack problem with series-parallel support. Operations Research Letters 30, 159–166 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ausiello, G., Boria, N., Giannakos, A., Lucarelli, G., Paschos, V.T.: Online maximum k-coverage. Technical Report 299, Université Paris-Dauphine, Cahiers du LAMSADE (2010)Google Scholar
  10. 10.
    Paschos, V.T.: A survey of approximately optimal solutions to some covering and packing problems. ACM Computing Surveys 29, 171–209 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Giorgio Ausiello
    • 1
  • Nicolas Boria
    • 2
  • Aristotelis Giannakos
    • 2
  • Giorgio Lucarelli
    • 2
  • Vangelis Th. Paschos
    • 2
    • 3
  1. 1.Dip. di Informatica e SistemisticaUniversità degli Studi di Roma “La Sapienza”Italy
  2. 2.LAMSADECNRS UMR 7243 and Université Paris-DauphineFrance
  3. 3.Institut Universitaire de FranceFrance

Personalised recommendations