New approximation algorithms for graph coloring
Journal of the ACM (JACM)
An Õ(n3/14)-coloring algorithm for 3-colorable graphs
Information Processing Letters
Approximate graph coloring by semidefinite programming
Journal of the ACM (JACM)
Approximating the independence number via the j -function
Mathematical Programming: Series A and B
The Lovász Theta Function and a Semidefinite Programming Relaxation of Vertex Cover
SIAM Journal on Discrete Mathematics
Zero knowledge and the chromatic number
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
How Good is the Goemans--Williamson MAX CUT Algorithm?
SIAM Journal on Computing
Constructing worst case instances for semidefinite programming based approximation algorithms
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Coloring k-colorable graphs using smaller palettes
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
On the integrality ratio of semidefinite relaxations of MAX CUT
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
On the Hardness of 4-Coloring a 3-Colorable Graph
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
The Lovász Number of Random Graphs
Combinatorics, Probability and Computing
On hard instances of approximate vertex cover
ACM Transactions on Algorithms (TALG)
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Sherali-adams relaxations of the matching polytope
Proceedings of the forty-first annual ACM symposium on Theory of computing
A better approximation ratio for the vertex cover problem
ACM Transactions on Algorithms (TALG)
Vertex cover resists SDPs tightened by local hypermetric inequalities
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
On the approximability of the vertex cover and related problems
AAIM'10 Proceedings of the 6th international conference on Algorithmic aspects in information and management
Integrality gaps of linear and semi-definite programming relaxations for Knapsack
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Strong and weak edges of a graph and linkages with the vertex cover problem
Discrete Applied Mathematics
Integrality Gaps of $2-o(1)$ for Vertex Cover SDPs in the Lovász-Schrijver Hierarchy
SIAM Journal on Computing
A better approximation ratio for the vertex cover problem
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
An edge-reduction algorithm for the vertex cover problem
Operations Research Letters
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We investigate the power of a strengthened SDP relaxation for graph coloring whose value is equal to a variant of the Lovász ϑ-function. We show families of graphs where the value of the relaxation is 2 + ε for any fixed ε 0, yet the chromatic number is nδ for some fixed δ 0, which is a function of ε. This demonstrates the bound provided by the SDP is not strong enough to color a 3-colorable graph with no(1) colors.Kleinberg and Goemans considered an SDP relaxation for vertex cover whose value is n - ϑ1/2 (ϑ1/2 being the variant of the ϑ-function introduced by Schrijver). They asked whether this is within a ratio of 2 - ε of the optimal vertex cover for any ε 0. Our construction answers this question negatively.