Polynomially Solvable Cases of Binary Quadratic Programs
We summarize in this chapter polynomially solvable subclasses of binary quadratic programming problems studied in the literature and report some new polynomially solvable subclasses revealed in our recent research. It is well known that the binary quadratic programming program is NP-hard in general. Identifying polynomially solvable subclasses of binary quadratic programming problems not only offers theoretical insight into the complicated nature of the problem but also provides platforms to design relaxation schemes for exact solution methods. We discuss and analyze in this chapter six polynomially solvable subclasses of binary quadratic programs, including problems with special structures in the matrix Q of the quadratic objective function, problems defined by a special graph or a logic circuit, and problems characterized by zero duality gap of the SDP relaxation. Examples and geometric illustrations are presented to provide algorithmic and intuitive insights into the problems.
Keywordsbinary quadratic programming polynomial solvability series-parallel graph logic circuit lagrangian dual SDP relaxation
This research was partially supported by the Research Grants Council of Hong Kong under grant 414207 and by the National Natural Science Foundation of China under grants 70671064 and 70832002.
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