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Bounded Wang tilings with integer programming and graph-based heuristics.

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This study introduces new methods for Wang tiling on bounded areas, crucial for materials engineering. Heuristic algorithms offer efficient solutions, outperforming traditional methods and revealing errors in existing aperiodic tile sets.

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Area of Science:

  • Computational Geometry
  • Discrete Mathematics
  • Materials Science

Background:

  • Wang tiles are used for pattern compression and avoiding periodicity.
  • Most research focuses on tiling infinite planes, limiting practical applications.
  • Emerging applications in materials engineering require bounded tiling solutions.

Purpose of the Study:

  • To develop methods for constructing valid Wang tilings on bounded regions.
  • To offer efficient algorithms for complex tiling problems.
  • To identify errors in established aperiodic tile sets.

Main Methods:

  • Four integer programming formulations (decision, maximum-rectangular tiling, maximum cover, maximum adjacency constraint satisfaction).
  • Extensions with tile-based, color-based, packing, and variable-sized periodic constraints.
  • An efficient heuristic algorithm based on shortest path search in directed acyclic graphs.

Main Results:

  • Integer programming formulations for bounded Wang tiling construction.
  • Heuristic algorithms provide competitive results significantly faster than exact methods.
  • Approximation guarantees of 1/2 and 2/3 for the heuristic algorithm.
  • Identification of errors in the Knuth and Lagae corner aperiodic tile sets.

Conclusions:

  • The developed methods enable efficient Wang tiling on bounded areas, relevant for materials engineering.
  • Heuristic algorithms offer a practical and fast alternative to exact methods for tiling problems.
  • This work corrects fundamental errors in known aperiodic tile sets, advancing the field.