inapproximability of the smallest superpolyomino problem
DESCRIPTION
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). - PowerPoint PPT PresentationTRANSCRIPT
Inapproximability of the Smallest Superpolyomino Problem
Andrew WinslowTufts University
Polyominoes
Colored poly-squares
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:
(stick)
Smallest superpolyomino problem is NP-hard.
But greedy 4-approximation exists!
Yields simple, useful string compression.
(stick)
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.
(stick)
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:
(stick)