New approximation algorithms for graph coloring

Blum, Avrim

doi:10.1145/176584.176586

Cited by 97 publications

(86 citation statements)

References 14 publications

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“…This semicoloring can be used to legally color G using O(n 0.387 ) colors by applying Lemma 5.1. The bound just described is (marginally) weaker than the guarantee of a O(n 0.375 ) coloring due to Blum [9]. We now improve this result by constructing a semicoloring with fewer colors.…”

Section: Wigderson's Algorithmmentioning

confidence: 82%

“…The main algorithm presented here has been derandomized in a recent work of Mahajan and Ramesh [35]. By combining our techniques with those of Blum [9], Blum and Karger [BK97] have given a 3-coloring algorithm with approximation ratioÕ(n 3/14 ).…”

Section: Discussionmentioning

confidence: 99%

“…Wigderson [42] improved this bound by giving an elegant algorithm that uses O(n 1−1/(k−1) ) colors to legally color a k-colorable graph. Subsequently, other polynomial time algorithms were provided by Blum [9] that use O(n 3/8 log 8/5 n) colors to legally color an n-vertex 3-colorable graph. This result generalizes to coloring a k-colorable graph with O(n 1−1/(k−4/3) log 8/5 n) colors.…”

Section: Introductionmentioning

confidence: 99%

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Approximate graph coloring by semidefinite programming

Karger

Motwani

Sudan

1998

J. ACM

299

144

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We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on n vertices with min{O(∆ 1/3 log 1/2 ∆ log n), O(n 1/4 log 1/2 n)} colors where ∆ is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first non-trivial approximation result as a function of the maximum degree ∆. This result can be generalized to k-colorable graphs to obtain a coloring using min{O(∆ 1−2/k log 1/2 ∆ log n), O(n 1−3/(k+1) log 1/2 n)} colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovász ϑ-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the ϑ-function. *

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Section: Wigderson's Algorithmmentioning

confidence: 82%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximate graph coloring by semidefinite programming

Karger

Motwani

Sudan

1998

J. ACM

299

144

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“…Finding such a k coloring is NP-hard for general graphs and it is even NPhard to find a constant approximation [12]. Thus, even if a graph is k-colorable, a good static assignment algorithm will probably not find such a coloring.…”

Section: Drawbacks Of Static Techniquesmentioning

confidence: 99%

Distributed channel management in uncoordinated wireless environments

Mishra

Shrivastava

Agrawal

et al. 2006

Proceedings of the 12th Annual International Conference on Mobile Computing and Networking

151

119

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Wireless 802.11 hotspots have grown in an uncoordinated fashion with highly variable deployment densities. Such uncoordinated deployments, coupled with the difficulty of implementing coordination protocols, has often led to conflicting configurations (e.g., in choice of transmission power and channel of operation) among the corresponding Access Points (APs). Overall, such conflicts cause both unpredictable network performance and unfairness among clients of neighboring hotspots. In this paper, we focus on the fairness problem for uncoordinated deployments. We study this problem from the channel assignment perspective. Our solution is based on the notion of channel-hopping, and meets all the important design considerations for control methods in uncoordinated deployments -distributed in nature, minimal to zero coordination among APs belonging to different hotspots, simple to implement, and interoperable with existing standards. In particular, we propose a specific algorithm called MAXchop, which works efficiently when using only non-overlapping wireless channels, but is particularly effective in exploiting partially-overlapped channels that have been proposed in recent literature. We also evaluate how our channel assignment approach complements previously proposed carrier sensing techniques in providing further performance improvements. Through extensive simulations on real hotspot topologies and evaluation of a full implementation of this technique, we demonstrate the efficacy of these techniques for not only fairness, but also the aggregate throughput, metrics.We believe that this is the first work that brings into focus the fairness properties of channel hopping techniques and we hope that the insights from this research will be applied to other domains where a fair division of a system's resources is an important consideration.

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“…On the other hand, an approximation algorithm for coloring an n-vertex graph G were proposed in [5] [17] [19]. Interested readers may consult [24] [5] [19] for more solution methods, exact and approximate, to the graph coloring problem.…”

Section: Groupingmentioning

confidence: 99%

Efficient Resource Allocation for IEEE 802.15.3(a) Ad Hoc Networks

Tseng

Shieh

et al. 2004

Intelligence in Communication Systems

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Abstract. IEEE 802.15.3(a) is working to design a higher speed PHY enhancement amendment to 802.15.3 such as ultra-wideband which can provide precise timing and location information. This paper proposes an effective resource allocation mechanism which can guarantee the QoS properties of multimedia traffics and provide higher channel utilization by the aid of location information. The proposed mechanism, which is executed by the piconet coordinator, consists of three components: grouping, call admission control, and best effort traffic maximization.The grouping component groups non-interfering traffics together so that they are allowed to be transmitted simultaneously. The problem of grouping is shown equivalent to the graph coloring problem for which a lot of algorithms are available. The CAC component is responsible for QoS guarantee. The best effort traffic maximization component attempts to maximize the total throughput of best effort traffics. The maximization problem is formulated as a linear programming. Finally, simulation results verify the ability of QoS guarantee and higher channel utilization.

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New approximation algorithms for graph coloring

Cited by 97 publications

References 14 publications

Approximate graph coloring by semidefinite programming

Approximate graph coloring by semidefinite programming

Distributed channel management in uncoordinated wireless environments

Efficient Resource Allocation for IEEE 802.15.3(a) Ad Hoc Networks

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