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Approximating the Maximum Sharing Problem

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Algorithms and Data Structures (WADS 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4619))

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Abstract

In the maximum sharing problem ( MS ), we want to compute a set of (non-simple) paths in an undirected bipartite graph covering as many nodes as possible of the first node layer of the graph, with the constraint that all paths have both endpoints in the second node layer and no node in that layer is covered more than once. MS is equivalent to the node-duplication based crossing elimination problem ( NDCE ) that arises in the design of molecular quantum-dot cellular automata (QCA) circuits and the physical synthesis of BDD based regular circuit structures in VLSI design. We show that MS is NP-hard, present a polynomial-time 1.5-approximation algorithm, and show that MS cannot be approximated with a factor better than \(740\over 739\) unless P = NP.

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References

  1. Antonelli, D.A., Chen, D.Z., Dysart, T.J., Hu, X.S., Khang, A.B., Kogge, P.M., Murphy, R.C., Niemier, M.T.: Quantum-dot cellular automata (QCA) circuit partitioning: problem modeling and solutions. In: DAC 2004. Proc. 41st ACM/IEEE Design Automation Conf., pp. 363–368. IEEE Computer Society Press, Los Alamitos (2004)

    Chapter  Google Scholar 

  2. Berman, P., Karpinski, M.: \(8 \over 7\)-approximation algorithm for (1,2)-TSP. In: SODA 2006. Proc. 17th Annual ACM-SIAM Symp. on Discrete Algorithms, pp. 641–648. ACM Press, New York (2006)

    Chapter  Google Scholar 

  3. Cao, A., Koh, C.-K.: Non-crossing ordered BDD for physical synthesis of regular circuit structure. In: Proc. International Workshop on Logic and Synthesis, pp. 200–206 (2003)

    Google Scholar 

  4. Chaudhary, A., Chen, D.Z., Hu, X.S., Niemier, M.T., Ravinchandran, R., Whitton, K.M.: Eliminating wire crossings for molecular quantum-dot cellular automata implementation. In: Proc. of IEEE/ACM International Conference on Computer-Aided Design, pp. 565–571. ACM Press, New York (2005)

    Google Scholar 

  5. Chen, D.Z., Fleischer, R., Li, J., Xie, Z., Zhu, H.: On approximating the maximum simple sharing problem. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 547–556. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  6. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Englewood Cliffs (1998)

    Google Scholar 

  7. Eades, P., Whitesides, S.: Drawing graphs in two layers. Theor. Comput. Sci. 131, 361–374 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  8. Eades, P., Wormald, N.C.: Edge crossings in drawings of bipartite graphs. Algorithmica 11(4), 379–403 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Edmonds, J.: Paths, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)

    MATH  MathSciNet  Google Scholar 

  10. Engebretsen, L., Karpinski, M.: TSP with bounded metrics. Journal of Computer and System Sciences 72(4), 509–546 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Finocchi, I.: Layered Drawings of Graphs with Crossing Constraints. In: Proc. 9th Annual International Computing and Combinatorics Conference, pp. 357–367 (2001)

    Google Scholar 

  12. Gabow, H.N.: Data Structures for Weighted Matching and Nearest Common Ancestors with Linking. In: SODA 1990. Proc. 7th Ann. ACM-SIAM Symp. on Discrete Algorithms, pp. 434–443. ACM Press, New York (1990)

    Google Scholar 

  13. Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM Journal on Algebraic and Discrete Methods 4(3), 312–316 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lengauer, T.: Combinatorial Algorithms for Integrated Circuit Layout. Wiley, Chichester (1990)

    MATH  Google Scholar 

  15. Papadimitriou, C.H., Yannakakis, M.: The traveling salesman problem with distances one and two. Mathematics of Operations Research 18(1), 1–11 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Tougaw, P.D., Lent, C.S.: Logical devices implemented using quantum cellular automata. J. of App. Phys. 75, 1818 (1994)

    Article  Google Scholar 

  17. Waterman, M.S., Griggs, J.R.: Interval graphs and maps of DNA. Bull. Math. Biol. 48(2), 189–195 (1986)

    MATH  MathSciNet  Google Scholar 

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Frank Dehne Jörg-Rüdiger Sack Norbert Zeh

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Chaudhary, A. et al. (2007). Approximating the Maximum Sharing Problem. In: Dehne, F., Sack, JR., Zeh, N. (eds) Algorithms and Data Structures. WADS 2007. Lecture Notes in Computer Science, vol 4619. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73951-7_6

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  • DOI: https://doi.org/10.1007/978-3-540-73951-7_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73948-7

  • Online ISBN: 978-3-540-73951-7

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