Inapproximability of the Smallest Superpolyomino Problem

Andrew WinslowTufts University

Polyominoes

Colored poly-squares

Rotation disallowed

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Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:

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Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:

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Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:

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Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:

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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 is NP-hard.

But greedy 4-approximation exists!

Yields simple, useful string compression.

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Known results

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:

(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)

Reduce from chromatic number.

Reduction Idea

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.

Elements

Sets

Smallest superpolyomino problem is NP-hard.

But greedy 4-approximation exists.

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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

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

Gives superpolyomino at most 4 times size of optimal: a 4-approximation.

output:

input:

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)

Cheating is as bad as worst cover.

So smallest superpolyomino is a good coverand finding it is NP-hard.

Smallest superpolyomino problem

Given a set of polyominoes:

Find a small superpolyomino:

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