Semidefinite programming studies optimization problems with a linear objective function over semidefinite constraints. It shares many interesting properties with linear programming. In particular, a semidefinite program can be solved in polynomial time. Moreover, an integer quadratic programcan be transformed into a semidefinite programthrough relaxation. Therefore, if a combinatorial optimization problem can be formulated as an integer quadratic program, then we can approximate it using the semidefinite programming relaxation and other related techniques such as the primal’dual schema. As the semidefinite programming relaxation is a higher-order relaxation, it often produces better results than the linear programming relaxation, even if the underlying problem can be formulated as an integer linear program. In this chapter, we introduce the fundamental concepts of semidefinite programming, and demonstrate its application to the approximation of NP-hard combinatorial optimization problems, with various rounding techniques.
KeywordsApproximation Algorithm Performance Ratio Cholesky Factorization Feasible Domain Linear Objective Function
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