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Inapproximability of the Smallest Superpolyomino Problem
Andrew WinslowTufts University
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Polyominoes
Colored poly-squares
Rotation disallowed
(stick)
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Smallest superpolyomino problem
Given a set of polyominoes:
Find a small superpolyomino:
(stick)
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Smallest superpolyomino problem
Given a set of polyominoes:
Find a small superpolyomino:
(stick)
![Page 5: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/5.jpg)
Smallest superpolyomino problem
Given a set of polyominoes:
Find a small superpolyomino:
(stick)
![Page 6: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/6.jpg)
Smallest superpolyomino problem
Given a set of polyominoes:
Find a small superpolyomino:
(stick)
![Page 7: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/7.jpg)
Smallest superpolyomino problem
Given a set of polyominoes:
Find a small superpolyomino:
(stick)
![Page 8: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/8.jpg)
Smallest superpolyomino problem
Given a set of polyominoes:
Find a small superpolyomino:
(stick)
![Page 9: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/9.jpg)
Smallest superpolyomino problem
Given a set of polyominoes:
Find a small superpolyomino:
(stick)
![Page 10: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/10.jpg)
Smallest superpolyomino problem is NP-hard.
But greedy 4-approximation exists!
Yields simple, useful string compression.
(stick)
Known results
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Smallest Superpolyomino Problem
Given a set of polyominoes:
Find a small superpolyomino:
![Page 12: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/12.jpg)
Smallest Superpolyomino Problem
Given a set of polyominoes:
Find a small superpolyomino:
![Page 13: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/13.jpg)
Smallest Superpolyomino Problem
Given a set of polyominoes:
Find a small superpolyomino:
![Page 14: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/14.jpg)
Smallest Superpolyomino Problem
Given a set of polyominoes:
Find a small superpolyomino:
![Page 15: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/15.jpg)
Smallest Superpolyomino Problem
Given a set of polyominoes:
Find a small superpolyomino:
![Page 16: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/16.jpg)
Smallest Superpolyomino Problem
Given a set of polyominoes:
Find a small superpolyomino:
![Page 17: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/17.jpg)
Smallest Superpolyomino Problem
Given a set of polyominoes:
Find a small superpolyomino:
![Page 18: Inapproximability of the Smallest Superpolyomino Problem](https://reader036.vdocuments.site/reader036/viewer/2022062811/56816209550346895dd234cd/html5/thumbnails/18.jpg)
Smallest Superpolyomino Problem
Given a set of polyominoes:
Find a small superpolyomino:
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(even if only two colors)O(n1/3 – ε)-approximation is NP-hard.
NP-hard even if only one color is used.
Simple, useful image compression? No
(ε > 0)
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Reduce from chromatic number.
Reduction Idea
Polyomino ≈ vertex.
Polyominoes can stack iff vertices aren’t adjacent.
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Generating polyominoes from input graph
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Chromatic number from superpolyomino
4 stacks ≈ 4-coloring
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Two-color polyomino sets
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One-color polyomino sets
Reduction from set cover.
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Elements
Sets
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Smallest superpolyomino problem is NP-hard. But greedy 4-approximation exists.
(stick)
Smallest superpolyomino problem is NP-hard. O(n1/3 – ε)-approximation is NP-hard.
One-color variant is NP-hard.
The good, the bad, and the inapproximable.
One-color variant is trivial. KNOWN
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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?
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Greedy approximation algorithm
Gives superpolyomino at most 4 times size of optimal: a 4-approximation.
output:
input:
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k is (n1-ε)-inapproximable.
Inapproximability ratio
So smallest superpolyomino is O(n1/3-ε)-inapproximable.
k-stack superpolyomino has size θ(k|V|2):
Stack size is θ(|V|2)
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Cheating is as bad as worst cover.
So smallest superpolyomino is a good coverand finding it is NP-hard.
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Smallest superpolyomino problem
Given a set of polyominoes:
Find a small superpolyomino:
(stick)