Loading in 5 sec....

Inapproximability of the Smallest Superpolyomino ProblemPowerPoint Presentation

Inapproximability of the Smallest Superpolyomino Problem

- By
**angie** - Follow User

- 81 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Inapproximability of the Smallest Superpolyomino Problem' - angie

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Smallest superpolyomino problem is NP-hard.

(stick)

But greedy 4-approximation exists!

Yields simple, useful string compression.

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.

- So smallest superpolyomino is a good cover

- and finding it is NP-hard.

Download Presentation

Connecting to Server..