Coloring k-colorable graphs using smaller palettes

1
Coloring k-colorable graphs using smaller
palettes

• Eran Halperin Ram Nathaniel Uri Zwick
• Tel Aviv University

2
New coloring results

• Coloring k-colorable graphs
• of maximum degree D using
• D1-2/k log1/kD colors
• (instead of D1-2/k log1/2D colors KMS)

3
New coloring results

• Coloring k-colorable graphs using
• na(k) colors (instead of nb(k) colors KMS)

4
An extension of Alon-Kahale

• AK If a graph contains an independent set of
  size n/km, k integer, then an independent set of
  size m3/(k1) can be found in polynomial time.
• Extension If a graph contains an independent set
  of size n/a, then an independent set of size
  nf(a) can be found in polynomial time, where

5
Graph coloring basics

• If in any k-colorable graph on n vertices we can
  find, in polynomial time, one of
• Two vertices that have the same color under some
  valid k-coloring
• An independent set of size W(n1-a)
• then we can color any k-colorable graph using
  O(na) colors.

6
Coloring 3-colorable graphs using O(n1/2) colors
Wigderson

• A graph with maximum degree D can be easily
  colored using D1 colors.
• If D lt n1/2, color using D1 colors.
• Otherwise, let v be a vertex of degree D.
• Then, N(v) is 2-colorable and contains an
  independent set of size D/2gt n1/2/2.

7
Vector k-Coloring KMS

• A vector k-coloring of a graph G(V,E) is a
  sequence of unit vectors v1,v2,,vn such that if
  (i,j) in E then ltvi,vjgt-1/(k-1).

8
Finding large independent sets

• Let G(V,E) be a 3-colorable graph.
• Let r be a random normally distributed vector in
  Rn. Let .
• I is obtained from I by removing a vertex from
  each edge of I.

9
Constructing the sets I and I
10
Analysis
11
Analysis (Cont.)
12
(No Transcript)
13
Analysis (Cont.)
14
Analysis (Cont.)
15
A simple observation
Suppose G(V,E) is k-colorable.

• Either GN(u,v) is (k-2)-colorable,
• or u and v get the same color under
• any a k-coloring of G.

16
A lemma of Blum

• Let G(V,E) be a k-colorable graph with
• minimum degree d
• for every
• Then, it is possible to construct, in polynomial
  time, a collection Ti of about n subsets of V
  such that at least one Ti satisfies
• TiW(d2/s)
• Ti has an independent subset of size

17
A lemma of Blum
18
Graph coloring techniques
Karger Motwani Sudan
Wigderson
Blum
Alon Kahale
Blum Karger
Our Algorithm
19
The new algorithm

• Step 0
• If k2, color the graph using 2 colors.
• If k3, color the graph using n3/14 colors using
  the algorithm of Blum and Karger.

20
The new algorithm

• Step 1
• Repeatedly remove from the graph vertices of
  degree at most na(k)/(1-2/k). Let U be the set of
  vertices removed, and WV-U.
• Average degree of GU is at most na(k)/(1-2/k).
• Minimum degree of GW at least na(k)/(1-2/k).
• If Ugtn/2, use KMS to find an independent set
  of size n/D1-2/k n1-a(k).

21
Step 1
Let dna(k)/(1-2/k).

• Average degree of GU is at most d.
• Minimum degree of GW at least d.

22
The new algorithm

• Step 2
• For every u,v such that N(u,v)gtn(1-a(k)/(1-a(k-2))
  ,
• apply the algorithm recursively on GN(u,v) and
  k-2.
• If GN(u,v) is (k-2)-colorable, we get an
  independent set of size N(u,v)1-a(k-2)gtn1-a(k).
• Otherwise, we can infer that u and v must be
  assigned the same color.

23
The new algorithm

• Step 3 If we reach this step then Wgtn/2, the
  minimum degree of GW is at least na(k)/(1-2/k),
• and for every u,v in W, N(u,v)gtn(1-a(k)/(1-a(k-2))
  .
• By Blums lemma, we can find a collection Ti of
  about n subsets of W such that at least one Ti
  satisfies TiW(d2/s) and Ti has an independent
  subset of size .
• By the extension of the Alon-Kahale result,
• we can find an IS of size

24
The recurrence relation
25
Hardness results

• It is NP-hard to 4-color 3-colorable graphs
  Khanna,Linial,Safra 93 Guruswami,Khanna 00
• For any k, it is NP-hard to k-color
• 2-colorable hypergraphs
• Guruswami,Hastad,Sudan 00