Coloring -colorable graphs using relatively small palettes

Halperin, Eran; Nathaniel, Ram; Zwick, Uri

doi:10.1016/s0196-6774(02)00217-1

Cited by 29 publications

(33 citation statements)

References 18 publications

(57 reference statements)

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“…The situation with coloring q-colorable graphs for q ≥ 4 is similar. The best known algorithm, due to Halperin et al [20], colors n vertex q-colorable graphs using O(n α q ) colors, where 0 < α q < 1 is some constant depending on q. For example, α 4 = A major ingredient in the above coloring algorithms is a semidefinite programming (SDP) relaxation of the chromatic number [21] called vector chromatic number, which we denote by χ v .…”

Section: Graph Coloringmentioning

confidence: 99%

“…Another strategy is to find the largest possible chromatic number q of a graph G satisfying minrk 2 (G) = 3, and to apply an algorithm for coloring q-colorable graphs to G. For example, by Inequality (2), χ(G) ≤ 8, and therefore the algorithm of [20] [29] a graph family G k (see Section 3) such that for any k, G k is the graph that has a maximum chromatic number among all the graphs whose complement has minrank k. That is, for any k,…”

Section: Algorithms For Linear Index Codingmentioning

confidence: 99%

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Linear Index Coding via Semidefinite Programming

Chlamtáč¹,

Haviv²

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef et al., FOCS, 2006).We show a polynomial time algorithm that, given an n vertex graph G with minrank k, finds a linear index code for G of length O(n f (k) ), where f (k) depends only on k. For example, for k = 3 we obtain f (3) ≈ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan (J. ACM, 1998) for graph coloring and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank.At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovász ϑ-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

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Section: Graph Coloringmentioning

confidence: 99%

Section: Algorithms For Linear Index Codingmentioning

confidence: 99%

Linear Index Coding via Semidefinite Programming

Chlamtáč¹,

Haviv²

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The best algorithm known for larger values of q is due to Halperin et al [9], improving on a previous result of Karger et al [11]. Their algorithm solves APPROXIMATE-COLORING (q, Q) for Q = n αq where 0 < α q < 1 is some function of q.…”

Section: Introductionmentioning

confidence: 97%

Conditional Hardness for Approximate Coloring

Dinur¹,

Mossel²,

Regev³

2009

SIAM J. Comput.

104

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We study the APPROXIMATE-COLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q. We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot's 2-to-1 conjecture [Khot'02]. For q = 3, we base our hardness result on a certain '⊲< shaped' variant of his conjecture.We also prove that the problem ALMOST-3-COLORING ε is hard for any constant ε > 0, assuming Khot's Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3-color all but a ε fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least ε.Our result is based on bounding various generalized noise-stability quantities using the invariance principle of Mossel et al [MOO'05].

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“…Their breakthrough result has been slightly improved since, see e.g. [16]. Inspired by the definition of ϑ C (G) [6] we introduce a related copositive program Θ C (G) which strengthens the Lovász theta number toward the chromatic number.…”

Section: Introductionmentioning

confidence: 99%

Copositive programming motivated bounds on the stability and the chromatic numbers

Dukanovic

Rendl

2008

Math. Program.

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The Lovász theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovász theta number toward the chromatic number of G, which is shown to be equal to the fractional chromatic number of G. Solving copositive programs is NP-hard. This motivates the study of tractable approximations of the copositive cone. We investigate the Parrilo hierarchy to approximate this cone and provide computational simplifications for the approximation of the chromatic number of vertex transitive graphs. We provide some computational results indicating that the Lovász theta number can be strengthened significantly toward the fractional chromatic number of G on some Hamming graphs.

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Coloring -colorable graphs using relatively small palettes

Cited by 29 publications

References 18 publications

Linear Index Coding via Semidefinite Programming

Linear Index Coding via Semidefinite Programming

Conditional Hardness for Approximate Coloring

Copositive programming motivated bounds on the stability and the chromatic numbers

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