Inapproximability of the Smallest Superpolyomino Problem

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Inapproximability of the Smallest Superpolyomino Problem

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Inapproximability of the Smallest Superpolyomino Problem

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Inapproximability of the Smallest Superpolyomino Problem

Andrew Winslow

Tufts University

Colored poly-squares

(stick)

Rotation disallowed

(stick)

Given a set of polyominoes:

Find a small superpolyomino:

(stick)

Given a set of polyominoes:

Find a small superpolyomino:

(stick)

Given a set of polyominoes:

Find a small superpolyomino:

(stick)

Given a set of polyominoes:

Find a small superpolyomino:

(stick)

Given a set of polyominoes:

Find a small superpolyomino:

(stick)

Given a set of polyominoes:

Find a small superpolyomino:

(stick)

Given a set of polyominoes:

Find a small superpolyomino:

Known results

(stick)

But greedy 4-approximation exists!

Yields simple, useful string compression.

Given a set of polyominoes:

Find a small superpolyomino:

Given a set of polyominoes:

Find a small superpolyomino:

Given a set of polyominoes:

Find a small superpolyomino:

Given a set of polyominoes:

Find a small superpolyomino:

Given a set of polyominoes:

Find a small superpolyomino:

Given a set of polyominoes:

Find a small superpolyomino:

Given a set of polyominoes:

Find a small superpolyomino:

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

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

Reduction from set cover.

Sets

Elements

The good, the bad, and the inapproximable.

(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.

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?

input:

output:

Givessuperpolyomino at most 4 times

size of optimal: a 4-approximation.

- 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)

Given a set of polyominoes:

Find a small superpolyomino: