Shakhar Smorodinsky

Courant Institute, New-York University (NYU)

On the Chromatic Number of Some Geometric Hypergraphs

Hypergraph Coloring (definition)

A Hypergraph H=(V,E)

: V 1,…,k is a proper coloring if no hyperedge is monochromatic

Chromatic number (H) = min #colors needed for proper coloring H

R={1,2,3,4}, H(R) = (R,E),

E = { {1}, {2}, {3}, {4},{1,2}, {2,4},{2,3}, {1,3}, {1,2,3} {2,3,4}, {3,4} }

Example:

1

2

3

4

Conflict-Free Colorings

A Hypergraph H=(V,E)

: V 1,…,k is a Conflict-Free coloring (CF) if every hyperedge contains some unique color

CF-chromatic number CF(H) = min #colors needed to CF-Color H

Motivation for CF-colorings

Frequency Assignment in cellular networks

1

1

2

Goal: Minimize the total number of frequencies

A CF-Coloring Framework for R

1. Find a proper coloring of R

2. Color regions in largest color class with 1 and remove them

1

11

3. Recurse on remaining regions

2

2

4

3

1

11

22

4

3

New Framework for CF-coloring

Summary

CF-coloring a finite family of regions R:

1. i =0

2. While (R ) do {

3. i i+1

4. Find a Proper Coloring of H(R) with ``few’’ colors

5. R’ largest color class of ; R’ i

6. R R \R’

}

Framework for CF-coloring (cont)1. i=0

2. While (R ) do {

3. i i+1

4. Find a Coloring of H(R) with ``few’’ colors

5. R’ largest color class of ; R’

i 6. R R \R’

}

Framework is correct!

In fact, maximal color of any hyperedge is unique

“maximal” color i

Another i

Framework for CF-coloring (cont)1. i=0

2. While (R ) do {

3. i i+1

4. Find a Coloring of H(R) with ``few’’ colors

5. R’ largest color class of ; R’

i 6. R R \R’

}

Framework is correct!

In fact, maximal color of any hyperedge is unique

“maximal” color i

Another i

i th iteration

Not monochromatic

Not discard at i’th iteration

New Framework (cont)

CF-coloring a finite family of regions R:

i =0

1. While (R ) do {

2. i i+1

3. Find a Coloring of H(R) with ``few’’ colors

4. R’ largest color class of ; R’ i

5. R R \R’

}

Key question: Can we

make use only ``few” colors?

1. D = finite family of discs. (H(D)) ≤ 4 (tight!)

In fact, equivalent to the Four-Color Theorem.

2. R: axis-parallel rectangles.

(H(R)) ≤ 8log |R|

Asymptotically tight!

[Pach,Tardos 05] provided matching lower bound.

3. R : Jordan regions with ``low’’ ``union complexity’’

Then (H(R)) is ``small’’ (patience….)

For example: c s.t. (H(pseudo-discs)) ≤ c

Our Results on Proper Colorings

Chromatic number of H(R):

Definition: Union Complexity1

2

4

Union complexity:= #vertices on boundary

Example: pseudo-discs

Thm:

R : Regions s.t. any n have union complexity bounded by

u(n) then (H(R)) = o(u(n)/n)

Coloring pseudo-discs

Thm [Kedem, Livne, Pach, Sharir 86]:

The complexity of the union of any n pseudo-discs is ≤ 6n-12

Hence, u(n)/n is a constant. By above Thm, its chromatic number is O(1)

How about axis-parallel rectangles?

Union complexity could be quadratic !!!

Coloring axis-parallel rectangles

For general case, apply divide and conquer

≤ 8 colors

Coloring axis-parallel rectangles

For general case, apply divide and conquer

Obtain Coloring with

8log n colors

Summary CF-coloring i =0

1. While (R ) do {

2. i i+1

3. Find a Coloring of H(R) with ``few’’ colors

4. R’ largest color class of

5. R R \R’

}u(n) (H(R)) CF(H(R))

O(n)

(pseudo discs, etc)

O(1) O(log n)

O(n1+)Convex ``fat’’ regions,

etc

O(n) O(n)

Applied to regions with union complexity u(n)

General: Works for any hypergraph

Brief History[Even, Lotker, Ron, Smorodinsky 03]

• Any n discs can be CF-colored with O(log n) colors. Tight!

• Finding optimal coloring is NP-HARD even for congruent discs. (approximation algorithms are provided)

• For pts w.r.t discs (or homothetics), O(log n) colors suffice. [Har-Peled, Smorodinsky 03]

• Randomized framework for ``nice’’ regions, relaxed colorings, higher dimensions, VC-dimension …

Brief History (cont)[Alon, Smorodinsky 05] O(log3 k) colors for n discs s.t.

each intersects at most k others.

(Algorithmic) Online version:

• [Fiat et al., 05] pts arrive online on a line. CF-color w.r.t intervals. O(log2 n) colors.

• [Chen 05] [Bar-Noy, Hillaris, Smorodinsky 05] O(log n) colors w.h.p

• [Kaplan, Sharir, 05] pts arrive online in the plane

CF-color w.r.t congruent discs. O(log3 n) colors w.h.p• [Chen 05] CF-color w.r.t congruent discs.• O(log n) colors w.h.p

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