Advertisement

Hard and Easy Instances of L-Tromino Tilings

  • Javier T. Akagi
  • Carlos F. Gaona
  • Fabricio Mendoza
  • Manjil P. Saikia
  • Marcos VillagraEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)

Abstract

In this work we study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it either remains NP-complete or has a polynomial time algorithm. First, we characterize the possibility of when an Aztec rectangle has an L-tromino tiling, and hence also an Aztec diamond; if an Aztec rectangle has an unknown number of defects or holes, however, the problem of deciding a tiling is NP-complete. Then, we study tilings of arbitrary regions where only \(180^\circ \) rotations of L-trominoes are available. For this particular case we show that deciding the existence of a tiling remains NP-complete; yet, if a region does not contain so-called “forbidden polyominoes” as subregions, then there exists a polynomial time algorithm for deciding a tiling.

Keywords

Polyomino tilings Tromino Efficient tilings NP-completeness Aztec rectangle Aztec diamond Claw-free graphs 

References

  1. 1.
    Chin, P., Grimaldi, R., Heubach, S.: Tiling with L’s and squares. J. Integer Sequences 10(2), 3 (2007)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Conway, J.H., Lagarias, J.C.: Tiling with polyominoes and combinatorial group theory. J. Comb. Theor. Ser. A 53(2), 183–208 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Demaine, E.D., Demaine, M.L.: Jigsaw puzzles, edge matching, and polyomino packing: connections and complexity. Graphs and Combinatorics 23(1), 195–208 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Elkies, N., Kuperberg, G., Larsen, M., Propp, J.: Alternating-sign matrices and domino tilings. I. J. Algebraic Comb. 1(2), 111–132 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Horiyama, T., Ito, T., Nakatsuka, K., Suzuki, A., Uehara, R.: Complexity of tiling a polygon with trominoes or bars. Discrete Comput. Geom. 58(3), 686–704 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Mills, W.H., Robbins, D.P., Rumsey, H.: Alternating sign matrices and descending plane partitions. J. Comb. Theor. Ser. A 34(3), 340–359 (1983)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Minty, G.J.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theor. Ser. B 28(3), 284–304 (1980)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Moore, C., Robson, J.M.: Hard tiling problems with simple tiles. Discrete Comput. Geom. 26(4), 573–590 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Nakamura, D., Tamura, A.: A revision of Minty’s algorithm for finding a maximum weighted stable set of a claw-free graph. J. Oper. Res. Soc. Japan 44(2), 194–204 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Saikia, M.P.: Enumeration of domino tilings of an aztec rectangle with boundary defects. Adv. Appl. Math. 89, 41–66 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sbihi, N.: Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile. Discrete Math. 29(1), 53–76 (1980)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Javier T. Akagi
    • 1
  • Carlos F. Gaona
    • 1
  • Fabricio Mendoza
    • 1
  • Manjil P. Saikia
    • 2
  • Marcos Villagra
    • 1
    Email author
  1. 1.Universidad Nacional de AsunciónSan LorenzoParaguay
  2. 2.Fakultät für MathematikUniversität WienViennaAustria

Personalised recommendations