Abstract
We study the classical quadratic knapsack problem (QKP) on special graph classes. In this case the quadratic terms of the objective function are present only for certain pairs of knapsack items. These pairs are represented by the edges of a graph G=(V,E) whose vertices represent the knapsack items. We show that QKP permits an FPTAS on graphs of bounded treewidth and a PTAS on planar graphs and more generally on H-minor free graphs. The latter result is shown by adopting a technique of Demaine et al. (2005). We will also show strong NP-hardness of QKP on graphs that are 3-book embeddable, a natural graph class that is related to planar graphs. In addition we will argue that the problem might have a bad approximability behaviour on all graph classes containing large cliques (under certain complexity assumption used for showing hardness results for the densest k-subgraph problem).
This research was funded by the Austrian Science Fund (FWF): P23829.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Arora, S., Manokaran, R., Moshkovitz, D., Weinstein, O.: Inapproximabilty of Densest k-Subgraph from Average Case Hardness. Technical report (2011)
Bernhart, F., Kainen, P.C.: The book thickness of a graph. Journal of Combinatorial Theory, Series B 27(3), 320–331 (1979)
Chen, D.Z., Fleischer, R., Li, J.: Densest k-subgraph approximation on intersection graphs. In: Jansen, K., Solis-Oba, R. (eds.) WAOA 2010. LNCS, vol. 6534, pp. 83–93. Springer, Heidelberg (2011)
Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Embedding graphs in books: A layout problem with applications to vlsi design. SIAM Journal on Algebraic Discrete Methods 8(1), 33–58 (1987)
Corneil, D.G., Perl, Y.: Clustering and domination in perfect graphs. Discrete Applied Mathematics 9(1), 27–39 (1984)
Demaine, E.D., Hajiaghayi, M.T., Kawarabayashi, K.: Algorithmic graph minor theory: Decomposition, approximation, and coloring. In: 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005, pp. 637–646 (2005)
Feige, U.: Relations between average case complexity and approximation complexity. In: STOC, pp. 534–543. ACM (2002)
Feige, U., Peleg, D., Kortsarz, G.: The Dense k-Subgraph Problem. Algorithmica 29(3), 410–421 (2001)
Kainen, P.C., Overbay, S.: Book embeddings of graphs and a theorem of whitney. Technical report (2003)
Keil, J.M., Brecht, T.B.: The complexity of clustering in planar graphs. J. Combinatorial Mathematics and Combinatorial Computing 9, 155–159 (1991)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer (2004)
Khot, S.: Ruling Out PTAS for Graph Min-Bisection, Dense k-Subgraph, and Bipartite Clique. SIAM J. Comput. 36(4), 1025–1071 (2006)
Lovász, L.: Graph minor theory. Bulletin of the American Mathematical Society 43(1), 75–86 (2006)
Moore, C., Robson, J.M.: Hard tiling problems with simple tiles. Dscrete & Computational Geometry 26(4), 573–590 (2001)
Overbay, S.: Graphs with small book thickness. Missouri Journal of Mathematical Sciences 19(2), 121–130 (2007)
Pferschy, U., Schauer, J.: The Knapsack Problem with Conflict Graphs. Journal of Graph Algorithms and Applications 13(2), 233–249 (2009)
Pisinger, D.: The quadratic knapsack problem - a survey. Discrete Applied Mathematics 155, 623–648 (2007)
Pisinger, D., Rasmussen, A.B., Sandvik, R.: Solution of large quadratic knapsack problems through aggressive reduction. INFORMS Journal on Computing 19, 280–290 (2007)
Rader Jr., D.J., Woeginger, G.J.: The quadratic 0-1 knapsack problem with series-parallel support. Operations Research Letters 30, 159–166 (2002)
Raghavendra, P., Steurer, D., Tulsiani, M.: Reductions Between Expansion Problems. Electronic Colloquium on Computational Complexity (ECCC) 17, 172 (2010)
Rose, D.J.: On simple characterizations of k-trees. Discrete Mathematics 7, 317–322 (1974)
Yannakakis, M.: Four pages are necessary and sufficient for planar graphs. In: Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, STOC 1986, New York, NY, USA, pp. 104–108. ACM (1986)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Pferschy, U., Schauer, J. (2014). Approximating the Quadratic Knapsack Problem on Special Graph Classes. In: Kaklamanis, C., Pruhs, K. (eds) Approximation and Online Algorithms. WAOA 2013. Lecture Notes in Computer Science, vol 8447. Springer, Cham. https://doi.org/10.1007/978-3-319-08001-7_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-08001-7_6
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08000-0
Online ISBN: 978-3-319-08001-7
eBook Packages: Computer ScienceComputer Science (R0)