# Approximation algorithms for Max-3-Cut and other problems via complex semidefinite programming

## 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, 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 ≡ 3 (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 + 3/4π2arccos2(-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) 162-164)). 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 ℐ-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 Andersson et al. in the more general case. © 2003 Elsevier Inc.