On Approximation Algorithms for Coloring <I>k</I>-Colorable Graphs

Xuzhen XIE; Takao ONO; Tomio HIRATA

On Approximation Algorithms for Coloring k-Colorable Graphs

Xuzhen XIE, Takao ONO, Tomio HIRATA

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Karger, Motwani and Sudan presented a graph coloring algorithm based on semidefinite programming, which colors any k-colorable graph with maximum degree Δ using (Δ^1-2/k) colors. This algorithm leads to an algorithm for k-colorable graph using (n ^1-3/(k+1)) colors. This improved Wigderson's algorithm, which uses O(n^1-1/(k-1)) colors, containing as a subroutine an algorithm using (Δ+1) colors for graphs with maximum degree Δ. It is easy to imagine that an algorithm which uses less colors in terms of Δ leads to an algorithm which uses less colors in terms of n. In this paper, we consider this influence assuming that we have an algorithm which uses (Δ ^1-x/k) colors for 2<x<3. Specifically, we will show that the algorithms of Karger et al., of Blum and Karger and of Halperin et al. can be improved under this assumption.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E86-A No.5 pp.1046-1051

Publication Date: 2003/05/01

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Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)

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Xuzhen XIE, Takao ONO, Tomio HIRATA, "On Approximation Algorithms for Coloring k-Colorable Graphs" in IEICE TRANSACTIONS on Fundamentals, vol. E86-A, no. 5, pp. 1046-1051, May 2003, doi: .
Abstract: Karger, Motwani and Sudan presented a graph coloring algorithm based on semidefinite programming, which colors any k-colorable graph with maximum degree Δ using (Δ^1-2/k) colors. This algorithm leads to an algorithm for k-colorable graph using (n ^1-3/(k+1)) colors. This improved Wigderson's algorithm, which uses O(n^1-1/(k-1)) colors, containing as a subroutine an algorithm using (Δ+1) colors for graphs with maximum degree Δ. It is easy to imagine that an algorithm which uses less colors in terms of Δ leads to an algorithm which uses less colors in terms of n. In this paper, we consider this influence assuming that we have an algorithm which uses (Δ ^1-x/k) colors for 2<x<3. Specifically, we will show that the algorithms of Karger et al., of Blum and Karger and of Halperin et al. can be improved under this assumption.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1587/e86-a_5_1046/_p

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@ARTICLE{e86-a_5_1046,
author={Xuzhen XIE, Takao ONO, Tomio HIRATA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={On Approximation Algorithms for Coloring k-Colorable Graphs},
year={2003},
volume={E86-A},
number={5},
pages={1046-1051},
abstract={Karger, Motwani and Sudan presented a graph coloring algorithm based on semidefinite programming, which colors any k-colorable graph with maximum degree Δ using (Δ^1-2/k) colors. This algorithm leads to an algorithm for k-colorable graph using (n ^1-3/(k+1)) colors. This improved Wigderson's algorithm, which uses O(n^1-1/(k-1)) colors, containing as a subroutine an algorithm using (Δ+1) colors for graphs with maximum degree Δ. It is easy to imagine that an algorithm which uses less colors in terms of Δ leads to an algorithm which uses less colors in terms of n. In this paper, we consider this influence assuming that we have an algorithm which uses (Δ ^1-x/k) colors for 2<x<3. Specifically, we will show that the algorithms of Karger et al., of Blum and Karger and of Halperin et al. can be improved under this assumption.},
keywords={},
doi={},
ISSN={},
month={May},}

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TY - JOUR
TI - On Approximation Algorithms for Coloring k-Colorable Graphs
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1046
EP - 1051
AU - Xuzhen XIE
AU - Takao ONO
AU - Tomio HIRATA
PY - 2003
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E86-A
IS - 5
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - May 2003
AB - Karger, Motwani and Sudan presented a graph coloring algorithm based on semidefinite programming, which colors any k-colorable graph with maximum degree Δ using (Δ^1-2/k) colors. This algorithm leads to an algorithm for k-colorable graph using (n ^1-3/(k+1)) colors. This improved Wigderson's algorithm, which uses O(n^1-1/(k-1)) colors, containing as a subroutine an algorithm using (Δ+1) colors for graphs with maximum degree Δ. It is easy to imagine that an algorithm which uses less colors in terms of Δ leads to an algorithm which uses less colors in terms of n. In this paper, we consider this influence assuming that we have an algorithm which uses (Δ ^1-x/k) colors for 2<x<3. Specifically, we will show that the algorithms of Karger et al., of Blum and Karger and of Halperin et al. can be improved under this assumption.
ER -