Barrier functions in interior point methods
Mathematics of Operations Research
Self-scaled barriers and interior-point methods for convex programming
Mathematics of Operations Research
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)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
A Geometric Approach to Betweenness
SIAM Journal on Discrete Mathematics
The Chebyshev Polynomials of a Matrix
SIAM Journal on Matrix Analysis and Applications
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)
Primal-Dual Interior-Point Methods for Self-Scaled Cones
SIAM Journal on Optimization
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)
Derandomized dimensionality reduction with applications
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Using Complex Semidefinite Programming for Approximating MAX E2-LIN3
APPROX '01/RANDOM '01 Proceedings of the 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 5th International Workshop on Randomization and Approximation Techniques in Computer Science: Approximation, Randomization and Combinatorial Optimization
A combinatorial algorithm for MAX CSP
Information Processing Letters
An approximation algorithm for scheduling two parallel machines with capacity constraints
Discrete Applied Mathematics
Approximation algorithms for MAX-3-CUT and other problems via complex semidefinite programming
Journal of Computer and System Sciences - STOC 2001
Is constraint satisfaction over two variables always easy?
Random Structures & Algorithms
Max k-cut and approximating the chromatic number of random graphs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
The capacitated max-k-cut problem
ICCSA'05 Proceedings of the 2005 international conference on Computational Science and Its Applications - Volume Part IV
Product rules in semidefinite programming
FCT'07 Proceedings of the 16th international conference on Fundamentals of Computation Theory
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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 optimal solutions to these problems. In this paper, we consider using the cube roots of unity, 1, ei2&pgr;/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 [8] to obtain near-optimal solutions to the problems. In particular, we consider the problem of maximizing the total weight of satisfied equations xu-xv≡cmod 3 and inequations xu-xv≢cmod 3 , where xu∈0,1,2 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 .79373-approximation algorithm. If the instance contains only inequations (as it does for MAX-3-CUT), we obtain a performance guarantee of 712+34p 2arccos2 -1/4≈.83601. This compares with proven performance guarantees of .800217 for MAX-3-CUT (by Frieze and Jerrum [7]) and 13+10-8 for the general problem (by Andersson, Engebretson, and Håstad [2]). It matches the guarantee of .836008 for MAX-3-CUT found independently by de Klerk, Pasechnik, and Warners [4]. We show that all these algorithms are in fact identical in the case of MAX-3-CUT.