# Who is the Hardest One of All? NP-Completeness

# Who is the Hardest One of All? NP-Completeness

There are problems that cannot be solved, including 3-SAT, graph coloring, and Hamiltonian path. Each of these problems has the remarkable ability to express all the others, or any other problem in NP. This remarkable property is known as NP-completeness. This chapter examines the concept of NP-completeness and considers examples of NP-complete problems, from coloring maps to evaluating integrals. First, it translates the witness-checking programs of WITNESS EXISTENCE into simpler mathematical objects: Boolean circuits and formulas. It then reduces WITNESS EXISTENCE to 3-SAT, one of the most basic constraint satisfaction problems. It also demonstrates that reductions work by thinking about maps from solutions to solutions, explores some additional problems whose NP-completeness is somewhat surprising, and shows that NP-completeness exists even in mathematics. The chapter concludes by proving that Hamiltonian path is NP-complete.

*Keywords:*
problems, 3-SAT, Hamiltonian path, NP-completeness, integrals, witness-checking programs, Boolean circuits, reductions, solutions, mathematics

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