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Statistical physics of hard combinatorial optimization: The vertex cover problem

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 them in solving other optimization problems.