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Mathematics of Operations Research
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Self-scaled barriers and interior-point methods for convex programming
Mathematics of Operations Research
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
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Linear systems in Jordan algebras and primal-dual interior-point algorithms
Journal of Computational and Applied Mathematics - Special issue: dedicated to William B. Gragg on the occasion of his 60th Birthday
Approximate graph coloring by semidefinite programming
Journal of the ACM (JACM)
Checking polynomial identities over any field: towards a derandomization?
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
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A Geometric Approach to Betweenness
SIAM Journal on Discrete Mathematics
The Chebyshev Polynomials of a Matrix
SIAM Journal on Matrix Analysis and Applications
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Derandomizing Approximation Algorithms Based on Semidefinite Programming
SIAM Journal on Computing
A new way of using semidefinite programming with applications to linear equations mod p
Journal of Algorithms
Convex quadratic and semidefinite programming relaxations in scheduling
Journal of the ACM (JACM)
Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Primal-Dual Interior-Point Methods for Self-Scaled Cones
SIAM Journal on Optimization
A Tight Characterization of NP with 3 Query PCPs
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Approximating the value of two power proof systems, with applications to MAX 2SAT and MAX DICUT
ISTCS '95 Proceedings of the 3rd Israel Symposium on the Theory of Computing Systems (ISTCS'95)
MAX k-CUT and approximating the chromatic number of random graphs
Random Structures & Algorithms
Approximation Algorithms for Max 3-Section Using Complex Semidefinite Programming Relaxation
COCOA '09 Proceedings of the 3rd International Conference on Combinatorial Optimization and Applications
Conflict Resolution in the Scheduling of Television Commercials
Operations Research
Design of phase codes for radar performance optimization with a similarity constraint
IEEE Transactions on Signal Processing
Fast SDP algorithms for constraint satisfaction problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Probabilistic analysis of the semidefinite relaxation detector in digital communications
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
The entropy rounding method in approximation algorithms
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
On approximating complex quadratic optimization problems via semidefinite programming relaxations
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
Towards a unified architecture for in-RDBMS analytics
SIGMOD '12 Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data
Semidefinite relaxations for mixed 0-1 second-order cone program
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
New NP-Hardness Results for 3-Coloring and 2-to-1 Label Cover
ACM Transactions on Computation Theory (TOCT)
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A number of recent papers on approximation algorithms have used the square roots of unity, - 1 and 1, to represent binary decision variables for problems in combinatorial optimization, and have relaxed these to unit vectors in real space using semidefinite programming in order to obtain near optimum solutions to these problems. In this paper, we consider using the cube roots of unity, 1, ei2π/3, and ei4π/3, to represent ternary decision variables for problems in combinatorial optimization. Here the natural relaxation is that of unit vectors in complex space. We use an extension of semidefinite programming to complex space to solve the natural relaxation, and use a natural extension of the random hyperplane technique introduced by the authors in Goemans and Williamson (J. ACM 42 (1995) 1115-1145) to obtain near-optimum solutions to the problems. In particular, we consider the problem of maximizing the total weight of satisfied equations xu - xv ≡ c (mod 3) and inequations xu - xv ≢ c (mod 3), where xu ∈ {0, 1, 2} for all u. This problem can be used to model the MAX-3-CUT problem and a directed variant we call MAX-3-DICUT. For the general problem, we obtain a 0.793733-approximation algorithm. If the instance contains only inequations (as it does for MAX-3-CUT), we obtain a performance guarantee of 7/12 + 3/4π2 arccos2(-1/4) - ε 0.836008. This compares with proven performance guarantees of 0.800217 for MAX-3-CUT (by Frieze and Jerrum (Algorithmica 18 (1997) 67-81) and 1/3 + 10-8 for the general problem (by Andersson et al. (J. Algorithms 39 (2001) 162-204)). It matches the guarantee of 0.836008 for MAX-3-CUT found independently by de Klerk et al. (On approximate graph colouring and Max-k-Cut algorithms based on the ℓ-function, Manuscript, October 2000). We show that all these algorithms are in fact equivalent in the case of MAX-3- CUT, and that our algorithm is the same as that of Anderson et al. in the general case.