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Coloring k -colorable graphs using smaller palettes

Coloring k -colorable graphs using smaller palettes. Eran Halperin Ram Nathaniel Uri Zwick Tel Aviv University. New coloring results. Coloring k -colorable graphs of maximum degree D using D 1 -2/k log 1/k D colors (instead of D 1 -2/k log 1/2 D colors [KMS] ).

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Coloring k -colorable graphs using smaller palettes

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  1. Coloring k-colorable graphsusing smaller palettes Eran Halperin Ram Nathaniel Uri Zwick Tel Aviv University

  2. New coloring results Coloring k-colorable graphs of maximum degree Dusing D1-2/klog1/kDcolors (instead of D1-2/klog1/2Dcolors [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/k+m, k integer, then an independent set of size m3/(k+1) 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 D+1colors. If D <n1/2, color using D+1colors. Otherwise, let v be a vertex of degree D. Then, N(v) is 2-colorable and contains an independent set of size D/2>=n1/2/2.

  7. Vector k-Coloring [KMS] Avector 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 <vi,vj>=-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. Analysis (Cont.)

  13. Analysis (Cont.)

  14. A simple observation Suppose G=(V,E) is k-colorable. Either G[N(u,v)] is (k-2)-colorable, or u and v get the same color under any a k-coloring of G.

  15. 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 Tisatisfies: |Ti|=W(d2/s) Tihas an independent subset of size

  16. A lemma of Blum

  17. Graph coloring techniques Karger Motwani Sudan Wigderson Blum Alon Kahale Blum Karger Our Algorithm

  18. The new algorithm Step 0: If k=2, color the graph using 2 colors. If k=3, color the graph using n3/14 colors using the algorithm of Blum and Karger.

  19. 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 W=V-U. Average degree of G[U] is at most na(k)/(1-2/k). Minimum degree of G[W] at least na(k)/(1-2/k). If |U|>n/2, use [KMS] to find an independent set of size n/D1-2/k= n1-a(k).

  20. Step 1 Let d=na(k)/(1-2/k). Average degree of G[U] is at most d. Minimum degree of G[W] at least d.

  21. The new algorithm Step 2: For every u,v such that N(u,v)>n(1-a(k)/(1-a(k-2)), apply the algorithm recursively on G[N(u,v)] and k-2. If G[N(u,v)] is (k-2)-colorable, we get an independent set of size |N(u,v)|1-a(k-2)>n1-a(k). Otherwise, we can infer*that u and v must be assigned the same color.

  22. The new algorithm Step 3:If we reach this step then |W|>n/2, the minimum degree of G[W] is at least na(k)/(1-2/k), and for every u,v in W, N(u,v)>n(1-a(k)/(1-a(k-2)). By Blum’s lemma, we can find a collection {Ti} of about n subsets of W such that at least one Ti satisfies |Ti|=W(d2/s) and Tihas an independent subset of size . By the extension of the Alon-Kahale result, we can find an IS of size

  23. The recurrence relation

  24. 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]

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