Inapproximability of the smallest superpolyomino problem
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Inapproximability of the Smallest Superpolyomino Problem. Andrew Winslow Tufts University. Polyominoes. Colored poly-squares . (stick). Rotation disallowed. (stick). Smallest superpolyomino problem. Given a set of polyominoes : Find a small superpolyomino :. (stick).

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Inapproximability of the Smallest Superpolyomino Problem

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Inapproximability of the Smallest Superpolyomino Problem

Andrew Winslow

Tufts University


Polyominoes

Colored poly-squares

(stick)

Rotation disallowed


(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


Known results

Smallest superpolyomino problem is NP-hard. 

(stick)

But greedy 4-approximation exists! 

Yields simple, useful string compression. 


Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


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


Reduction Idea

Reduce from chromatic number.

Polyomino ≈vertex.

Polyominoes can stack iff

vertices aren’t adjacent.


Generating polyominoes from input graph


Chromatic number from superpolyomino

4 stacks ≈ 4-coloring


Two-color polyomino sets


One-color polyomino sets

Reduction from set cover.


Sets

Elements


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. 


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?


Greedy approximation algorithm

input:

output:

Givessuperpolyomino at most 4 times

size of optimal: a 4-approximation.


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.


  • Cheating is as bad as worst cover.

  • So smallest superpolyomino is a good cover

  • and finding it is NP-hard.


(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


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