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Inapproximability of the Smallest Superpolyomino Problem

This paper explores the complexities of the Smallest Superpolyomino Problem, wherein the objective is to find a minimal superpolyomino covering a given set of polyominoes. We establish that this problem is NP-hard, even under constraints such as limited colors. Nonetheless, we introduce a greedy 4-approximation algorithm that yields a superpolyomino at most four times the size of the optimal solution. In addition, we discuss various aspects related to approximation ratios, reduction ideas, and implications for string and image compression.

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Inapproximability of the Smallest Superpolyomino Problem

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  1. Inapproximability of the Smallest Superpolyomino Problem Andrew Winslow Tufts University

  2. Polyominoes Colored poly-squares (stick) Rotation disallowed

  3. (stick) Smallest superpolyomino problem Given a set of polyominoes: Find a small superpolyomino:

  4. (stick) Smallest superpolyomino problem Given a set of polyominoes: Find a small superpolyomino:

  5. (stick) Smallest superpolyomino problem Given a set of polyominoes: Find a small superpolyomino:

  6. (stick) Smallest superpolyomino problem Given a set of polyominoes: Find a small superpolyomino:

  7. (stick) Smallest superpolyomino problem Given a set of polyominoes: Find a small superpolyomino:

  8. (stick) Smallest superpolyomino problem Given a set of polyominoes: Find a small superpolyomino:

  9. (stick) Smallest superpolyomino problem Given a set of polyominoes: Find a small superpolyomino:

  10. Known results Smallest superpolyomino problem is NP-hard.  (stick) But greedy 4-approximation exists!  Yields simple, useful string compression. 

  11. Smallest Superpolyomino Problem Given a set of polyominoes: Find a small superpolyomino:

  12. Smallest Superpolyomino Problem Given a set of polyominoes: Find a small superpolyomino:

  13. Smallest Superpolyomino Problem Given a set of polyominoes: Find a small superpolyomino:

  14. Smallest Superpolyomino Problem Given a set of polyominoes: Find a small superpolyomino:

  15. Smallest Superpolyomino Problem Given a set of polyominoes: Find a small superpolyomino:

  16. Smallest Superpolyomino Problem Given a set of polyominoes: Find a small superpolyomino:

  17. Smallest Superpolyomino Problem Given a set of polyominoes: Find a small superpolyomino:

  18. Smallest Superpolyomino Problem Given a set of polyominoes: Find a small superpolyomino:

  19. O(n1/3 – ε)-approximation is NP-hard.  (ε > 0) (even if only two colors) NP-hard even if only one color is used.  Simple, useful image compression? No

  20. Reduction Idea Reduce from chromatic number. Polyomino ≈vertex. Polyominoes can stack iff vertices aren’t adjacent.

  21. Generating polyominoes from input graph

  22. Chromatic number from superpolyomino 4 stacks ≈ 4-coloring

  23. Two-color polyomino sets

  24. One-color polyomino sets Reduction from set cover.

  25. Sets Elements

  26. The good, the bad, and the inapproximable. Smallest superpolyomino problem is NP-hard.  (stick) KNOWN But greedy 4-approximation exists.  One-color variant is trivial.  Smallest superpolyomino problem is NP-hard.  • O(n1/3 – ε)-approximation is NP-hard.  • One-color variant is NP-hard. 

  27. Open(?) related problem The one-color variant is a constrained version of: “Given a set of polygons, find the minimum-area union of these polygons.” What is known? References?

  28. Greedy approximation algorithm input: output: Givessuperpolyomino at most 4 times size of optimal: a 4-approximation.

  29. Inapproximability ratio • Stack size is θ(|V|2) So smallest superpolyomino is O(n1/3-ε)-inapproximable. k-stack superpolyomino has size θ(k|V|2): • k is (n1-ε)-inapproximable.

  30. Cheating is as bad as worst cover. • So smallest superpolyomino is a good cover • and finding it is NP-hard.

  31. (stick) Smallest superpolyomino problem Given a set of polyominoes: Find a small superpolyomino:

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