658 Michel

We describe a few applications of semideenite programming in combinatorial optimization. Semideenite programming is a special case of convex programming where the feasible region is an aane subspace of the cone of positive semideenite matrices. There has been much interest in this area lately, partly because of applications in com-binatorial optimization and in control theory and also because of the development of eecient interior-point algorithms. The use of semideenite programming in combinatorial optimization is not new though. Eigenvalue bounds have been proposed for combinatorial optimization problems since the late 60's, see for example the comprehensive survey by Mohar and Poljak 20]. These eigenvalue bounds can often be recast as semideenite programs 1]. This reformulation is useful since it allows to exploit properties of convex programming such as duality and polynomial-time solvability, and it avoids the pitfalls of eigenvalue optimization such as non-diierentiability. An explicit use of semideenite programming in combinatorial optimization appeared in the seminal work of Lovv asz 16] on the so-called theta function, and this lead Grr otschel, Lovv asz and Schrijver 9, 11] to develop the only known (and non-combinatorial) polynomial-time algorithm to solve the maximum stable set problem for perfect graphs. In this paper, we describe a few applications of semideenite programming in combinatorial optimization. Because of space limitations, we restrict our attention to the Lovv asz theta function, the maximum cut problem 8], and the automatic generation of valid inequalities a la Lovv asz-Schrijver 17, 18]. This survey is much inspired by another (longer) survey written by the author 7]. However, new results on the power and limitations of the Lovv asz-Schrijver procedure are presented as well as a study of the maximum cut relaxation for graphs arising from association schemes.