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Generating Algebraic Expressions for Labeled Grid Graphs

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Combinatorial Optimization and Applications (COCOA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11346))

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Abstract

The paper investigates relationship between algebraic expressions and labeled graphs. We consider directed grid graphs having m rows and n columns. Our intent is to simplify the expressions of these graphs. With that end in view, we describe two methods which generate expressions for directed grid graphs. For both methods, lengths of the expressions grow polynomially with n while m is determined as a constant parameter. Besides, we apply these methods to a square grid graph in which the number of rows is equal to the number of columns. We prove that the lengths of the expressions derived by the methods depend exponentially and quasi-polynomially, respectively, on the size of the graph.

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References

  1. Ball, W.W.R., Coxeter, H.S.M.: Mathematical Recreations and Essays, 13th edn. The Macmillan Company, Dover, New York (1987)

    Google Scholar 

  2. Bar-Noy, A., Cheilaris, P., Lampis, M., Mitsou, V., Zachos, S.: Ordered coloring grids and related graphs. In: Kutten, S., Žerovnik, J. (eds.) SIROCCO 2009. LNCS, vol. 5869, pp. 30–43. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11476-2_4

    Chapter  MATH  Google Scholar 

  3. Bein, W.W., Brucker, P.: Greedy concepts for network flow problems. Discret. Appl. Math. 15(2–3), 135–144 (1986)

    Article  MathSciNet  Google Scholar 

  4. Bein, W.W., Kamburowski, J., Stallmann, M.F.M.: Optimal reduction of two-terminal directed acyclic graphs. SIAM J. Comput. 21(6), 1112–1129 (1992)

    Article  MathSciNet  Google Scholar 

  5. Duffin, R.J.: Topology of series-parallel networks. J. Math. Anal. Appl. 10, 303–318 (1965)

    Article  MathSciNet  Google Scholar 

  6. Finta, L., Liu, Z., Milis, I., Bampis, E.: Scheduling UET-UCT series-parallel graphs on two processors. Theor. Comput. Sci. 162(2), 323–340 (1996)

    Article  MathSciNet  Google Scholar 

  7. Golumbic, M.C., Gurvich, V.: Read-once functions. In: Crama, Y., Hammer, P.L. (eds.) Boolean Functions: Theory, Algorithms and Applications, pp. 519–560. Cambridge University Press, New York (2011)

    Google Scholar 

  8. Golumbic, M.C., Mintz, A., Rotics, U.: Factoring and recognition of read-once functions using cographs and normality and the readability of functions associated with partial k-trees. Discret. Appl. Math. 154(10), 1465–1477 (2006)

    Article  MathSciNet  Google Scholar 

  9. Golumbic, M.C., Perl, Y.: Generalized Fibonacci maximum path graphs. Discret. Math. 28, 237–245 (1979)

    Article  MathSciNet  Google Scholar 

  10. Grötschel, M., Martin, A., Weismantel, R.: Routing in grid graphs by cutting planes. ZOR - Math. Methods Oper. Res. 41(3), 255–275 (1995)

    Article  MathSciNet  Google Scholar 

  11. Hosoya, H., Harary, F.: On the matching properties of three fence graphs. J. Math. Chem. 12, 211–218 (1993)

    Article  MathSciNet  Google Scholar 

  12. Irani, S.: Coloring inductive graphs on-line. Algorithmica 11(1), 53–72 (1994)

    Article  MathSciNet  Google Scholar 

  13. Kaufmann, M., Gao, S., Thulasiraman, K.: On Steiner minimal trees in grid graphs and its application to VLSI routing. In: Du, D.-Z., Zhang, X.-S. (eds.) ISAAC 1994. LNCS, vol. 834, pp. 351–359. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-58325-4_199

    Chapter  MATH  Google Scholar 

  14. Kaufmann, M., Mehlhorn, K.: A linear-time algorithm for the homotopic routing problem in grid graphs. SIAM J. Comput. 23(2), 227–246 (1998)

    Article  MathSciNet  Google Scholar 

  15. Kirousis, L.M., Thilikos, D.M.: The linkage of a graph. SIAM J. Comput. 25(3), 626–647 (1996)

    Article  MathSciNet  Google Scholar 

  16. Korenblit, M.: Decomposition methods for generating algebraic expressions of full square rhomboids and other graphs. Discret. Appl. Math. 228, 60–72 (2017)

    Article  MathSciNet  Google Scholar 

  17. Korenblit, M., Levit, V.E.: On algebraic expressions of series-parallel and Fibonacci graphs. In: Calude, C.S., Dinneen, M.J., Vajnovszki, V. (eds.) DMTCS 2003. LNCS, vol. 2731, pp. 215–224. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-45066-1_17

    Chapter  MATH  Google Scholar 

  18. Krause, A.: Bounded Treewidth Graphs - A Survey German Russian Winter School, St. Petersburg, Russia (2003). http://wwwmayr.in.tum.de/konferenzen/Jass03/presentations/krause.pdf

  19. Levit, V.E., Korenblit, M.: A symbolic approach to scheduling of robotic lines. In: Intelligent Scheduling of Robots and Flexible Manufacturing Systems. The Center for Technological Education Holon, Israel, pp. 113–125 (1996)

    Google Scholar 

  20. Mintz, A., Golumbic, M.C.: Factoring Boolean functions using graph partitioning. Discret. Appl. Math. 149(1–3), 131–153 (2005)

    Article  MathSciNet  Google Scholar 

  21. Monma, C.L.: Sequencing with series-parallel precedence constraints. Math. Oper. Res. 4(3), 215–224 (1979)

    Article  MathSciNet  Google Scholar 

  22. Mundici, D.: Solution of Rota’s problem on the order of series-parallel networks. Adv. Appl. Math. 12, 455–463 (1991)

    Article  MathSciNet  Google Scholar 

  23. Noy, M., Ribó, A.: Recursively constructible families of graphs. Adv. Appl. Math. 32, 350–363 (2004)

    Article  MathSciNet  Google Scholar 

  24. Oikawa, M.K., Ferreira, J.E., Malkowski, S., Pu, C.: Towards algorithmic generation of business processes: from business step dependencies to process algebra expressions. In: Dayal, U., Eder, J., Koehler, J., Reijers, H.A. (eds.) BPM 2009. LNCS, vol. 5701, pp. 80–96. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03848-8_7

    Chapter  Google Scholar 

  25. Satyanarayana, A., Wood, R.K.: A linear time algorithm for computing K-terminal reliability in series-parallel networks. SIAM J. Comput. 14(4), 818–832 (1985)

    Article  MathSciNet  Google Scholar 

  26. Savicky, P., Woods, A.R.: The number of Boolean functions computed by formulas of a given size. Rand. Struct. Algorithms 13, 349–382 (1998)

    Article  MathSciNet  Google Scholar 

  27. Schmidt, J.P.: All highest scoring paths in weighted grid graphs and their application to finding all approximate repeats in strings. SIAM J. Comput. 27(4), 972–992 (1998)

    Article  MathSciNet  Google Scholar 

  28. Sesh Kumar, K.S.: Convex relaxations for learning bounded-treewidth decomposable graphs. In: Proceedings of 30th International Conference on Machine Learning (ICML2013), JMLR: W&CP, vol. 28 (2013)

    Google Scholar 

  29. Tamir, A.: A strongly polynomial algorithm for minimum convex separable quadratic cost flow problems on two-terminal series-parallel networks. Math. Program. 59, 117–132 (1993)

    Article  MathSciNet  Google Scholar 

  30. Wald, J.A., Colbourn, C.J.: Steiner trees in probabilistic networks. Microelectron. Reliabil. 23(5), 837–840 (1983)

    Article  Google Scholar 

  31. Wang, A.R.R.: Algorithms for multilevel logic optimization. Ph.D. thesis, University of California, Berkeley (1989)

    Google Scholar 

  32. Weisstein, E.W.: Grid Graph From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/GridGraph.html

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Authors and Affiliations

  1. Holon Institute of Technology, Holon, Israel

    Mark Korenblit

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  1. Mark Korenblit
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Correspondence to Mark Korenblit .

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Editors and Affiliations

  1. Department of Computer Science, Kennesaw State University, Marietta, GA, USA

    Donghyun Kim

  2. Department of Mathematics and Computer Science, North Carolina Central University, Durham, NC, USA

    R. N. Uma

  3. Department of Computer Science, Georgia State University, Atlanta, GA, USA

    Alexander Zelikovsky

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Korenblit, M. (2018). Generating Algebraic Expressions for Labeled Grid Graphs. In: Kim, D., Uma, R., Zelikovsky, A. (eds) Combinatorial Optimization and Applications. COCOA 2018. Lecture Notes in Computer Science(), vol 11346. Springer, Cham. https://doi.org/10.1007/978-3-030-04651-4_10

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  • DOI: https://doi.org/10.1007/978-3-030-04651-4_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-04650-7

  • Online ISBN: 978-3-030-04651-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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