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

Inapproximability of the Smallest Superpolyomino Problem

Andrew Winslow

Tufts University


Polyominoes

Polyominoes

Colored poly-squares

(stick)

Rotation disallowed


Smallest superpolyomino problem

(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem1

(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem2

(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem3

(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem4

(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem5

(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem6

(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem is np hard

Known results

Smallest superpolyomino problem is NP-hard. 

(stick)

But greedy 4-approximation exists! 

Yields simple, useful string compression. 


Smallest superpolyomino problem7

Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem8

Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem9

Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem10

Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem11

Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem12

Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem13

Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Smallest superpolyomino problem14

Smallest Superpolyomino Problem

Given a set of polyominoes:

Find a small superpolyomino:


Inapproximability of the smallest superpolyomino problem

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

Reduction Idea

Reduce from chromatic number.

Polyomino ≈vertex.

Polyominoes can stack iff

vertices aren’t adjacent.


Inapproximability of the smallest superpolyomino problem

Generating polyominoes from input graph


Inapproximability of the smallest superpolyomino problem

Chromatic number from superpolyomino

4 stacks ≈ 4-coloring


Two color polyomino sets

Two-color polyomino sets


One color polyomino sets

One-color polyomino sets

Reduction from set cover.


Inapproximability of the smallest superpolyomino problem

Sets

Elements


Smallest superpolyomino problem is np hard1

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

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

Greedy approximation algorithm

input:

output:

Givessuperpolyomino at most 4 times

size of optimal: a 4-approximation.


Inapproximability ratio

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.


Inapproximability of the smallest superpolyomino problem

  • Cheating is as bad as worst cover.

  • So smallest superpolyomino is a good cover

  • and finding it is NP-hard.


Smallest superpolyomino problem15

(stick)

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:


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