Data Structures and Algorithm Analysis - Computer Science at ...

Sec. 17.2 Hard Problems 547for TRAVELING SALESMAN and all related problems. Because TRAVELINGSALESMAN is known to be N P-complete, if a polynomial time algorithm were tobe found for this problem, then all problems in N P would also be solvable in polynomialtime. Conversely, if we were able to prove that TRAVELING SALESMANhas an exponential time lower bound, then we would know that P ≠ N P.17.2.2 N P-Completeness ProofsTo start the process of being able to prove problems are N P-complete, we need toprove just one problem H is N P-complete. After that, to show that any problemX is N P-hard, we just need to reduce H to X. When doing N P-completenessproofs, it is very important not to get this reduction backwards! If we reduce candidateproblem X to known hard problem H, this means that we use H as a step tosolving X. All that means is that we have found a (known) hard way to solve X.However, when we reduce known hard problem H to candidate problem X, thatmeans we are using X as a step to solve H. And if we know that H is hard, thatmeans X must also be hard (because if X were not hard, then neither would H behard).So a crucial first step to getting this whole theory off the ground is finding oneproblem that is N P-hard. The first proof that a problem is N P-hard (and becauseit is in N P, therefore N P-complete) was done by Stephen Cook. For this feat,Cook won the first Turing award, which is the closest Computer Science equivalentto the Nobel Prize. The “grand-daddy” N P-complete problem that Cook used iscall SATISFIABILITY (or SAT for short).A Boolean expression includes Boolean variables combined using the operatorsAND (·), OR (+), and NOT (to negate Boolean variable x we write x). Aliteral is a Boolean variable or its negation. A clause is one or more literals OR’edtogether. Let E be a Boolean expression over variables x 1 , x 2 , ..., x n . Then wedefine Conjunctive Normal Form (CNF) to be a Boolean expression written as aseries of clauses that are AND’ed together. For example,E = (x 5 + x 7 + x 8 + x 10 ) · (x 2 + x 3 ) · (x 1 + x 3 + x 6 )is in CNF, and has three clauses. Now we can define the problem SAT.SATISFIABILITY (SAT)Input: A Boolean expression E over variables x 1 , x 2 , ... in Conjunctive NormalForm.Output: YES if there is an assignment to the variables that makes E true,NO otherwise.Cook proved that SAT is N P-hard. Explaining Cook’s proof is beyond thescope of this book. But we can briefly summarize it as follows. Any decision