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• W4230348051 abstract "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, e i 2 π /3 , and e i 4 π /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 x u −x v ≡c ( mod 3) and inequations x u −x v ≢c ( mod 3) , where x u ∈{0,1,2} for all u . This problem can be used to model the M ax -3-C ut problem and a directed variant we call M ax -3-D icut . For the general problem, we obtain a 0.793733-approximation algorithm. If the instance contains only inequations (as it does for M ax -3-C ut ), we obtain a performance guarantee of 7 12 + 3 4π 2 arccos 2 (−1/4)−ε>0.836008 . This compares with proven performance guarantees of 0.800217 for M ax -3-C ut (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 M ax -3-C ut found independently by de Klerk et al. (On approximate graph colouring and Max- k -Cut algorithms based on the ϑ -function, Manuscript, October 2000). We show that all these algorithms are in fact equivalent in the case of M ax -3-C ut , and that our algorithm is the same as that of Andersson et al. in the more general case." @default.
• W4230348051 created "2022-05-11" @default.
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• W4230348051 date "2004-03-01" @default.
• W4230348051 modified "2023-10-16" @default.
• W4230348051 title "Approximation algorithms for Max-3-Cut and other problems via complex semidefinite programming" @default.
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• W4230348051 doi "https://doi.org/10.1016/j.jcss.2003.07.012" @default.
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