The 1-Neighbour Knapsack Problem

  • Glencora Borradaile
  • Brent Heeringa
  • Gordon Wilfong
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)


We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. We give approximation algorithms and hardness results when the nodes have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.


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  1. 1.
    Boland, N., Fricke, C., Froyland, G., Sotirov, R.: Clique-based facets for the precedence constrained knapsack problem. Technical report. Tilburg University Repository, Netherlands (2005),
  2. 2.
    Borradaile, G., Heeringa, B., Wilfong, G.: The knapsack problem with neighbour constraints. CoRR, abs/0910.0777 (2011)Google Scholar
  3. 3.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Goundan, P.R., Schulz, A.S.: Revisiting the greedy approach to submodular set function maximization (2009) (preprint)Google Scholar
  5. 5.
    Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: Proceedings of STOC, pp. 585–594 (2003)Google Scholar
  6. 6.
    Ibarra, O.H., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM 22, 463–468 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Johnson, D.S., Niemi, K.A.: On knapsacks, partitions, and a new dynamic programming technique for trees. Mathematics of Operations Research, 1–14 (1983)Google Scholar
  8. 8.
    Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Heidelberg (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Khuller, S., Moss, A., Naor(Seffi), J.: The budgeted maximum coverage problem. Inf. Process. Lett. 70(1), 39–45 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kolliopoulos, S.G., Steiner, G.: Partially ordered knapsack and applications to scheduling. Discrete Applied Mathematics 155(8), 889–897 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kulik, A., Shachnai, H., Tamir, T.: Maximizing submodular set functions subject to multiple linear constraints. In: Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, pp. 545–554. Society for Industrial and Applied Mathematics, Philadelphia (2009)CrossRefGoogle Scholar
  12. 12.
    Lee, J., Mirrokni, V.S., Nagarajan, V., Sviridenko, M.: Non-monotone submodular maximization under matroid and knapsack constraints. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pp. 323–332. ACM, New York (2009)Google Scholar
  13. 13.
    Sviridenko, M.: A note on maximizing a submodular set function subject to a knapsack constraint. Operations Research Letters 32(1), 41–43 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Vazirani, V.: Approximation Algorithms. Springer, Berlin (2001)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Glencora Borradaile
    • 1
  • Brent Heeringa
    • 2
  • Gordon Wilfong
    • 3
  1. 1.Oregon State UniversityUnited States
  2. 2.Williams CollegeUnited Kingdom
  3. 3.Bell LabsUnited States

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