Zhou, HJ (reprint author), Chinese Acad Sci, Inst Theoret Phys, State Key Lab Theoret Phys, Beijing 100190, Peoples R China.
; Zhou Hai-Jun] Chinese Acad Sci, Inst Theoret Phys, State Key Lab Theoret Phys, Beijing 100190, Peoples R China
Typical-case computation complexity is a research topic at the boundary of computer science, applied mathematics, and statistical physics. In the last twenty years, the replica-symmetry-breaking mean field theory of spin glasses and the associated message-passing algorithms have greatly deepened our understanding of typical-case computation complexity. In this paper, we use the vertex cover problem, a basic nondeterministic-polynomial (NP)-complete combinatorial optimization problem of wide application, as an example to introduce the statistical physical methods and algorithms. We do not go into the technical details but emphasize mainly the intuitive physical meanings of the message-passing equations. A nonfamiliar reader shall be able to understand to a large extent the physics behind the mean field approaches and to adjust the mean field methods in solving other optimization problems.