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S ATISFIABILITY, or SAT (recall Exercise 328 and Section 53), is a problem of great practical importance, with applications ranging from chip testing and computer design to image analysis and software engineering It is also a canonical hard problem Here s what an instance of SAT looks like: (x y z) (x y) (y z) (z x) (x y z)
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This is a Boolean formula in conjunctive normal form (CNF) It is a collection of clauses (the parentheses), each consisting of the disjunction (logical or, denoted ) of several literals, where a literal is either a Boolean variable (such as x) or the negation of one (such as x) A satisfying truth assignment is an assignment of false or true to each variable so that every clause contains a literal whose value is true The SAT problem is the following: given a Boolean formula in conjunctive normal form, either nd a satisfying truth assignment or else report that none exists In the instance shown previously, setting all variables to true, for example, satis es every clause except the last Is there a truth assignment that satis es all clauses With a little thought, it is not hard to argue that in this particular case no such truth assignment exists (Hint: The three middle clauses constrain all three variables to have the same value) But how do we decide this in general Of course, we can always search through all truth assignments, one by one, but for formulas with n variables, the number of possible assignments is exponential, 2n S AT is a typical search problem We are given an instance I (that is, some input data specifying the problem at hand, in this case a Boolean formula in conjunctive normal form), and we are asked to nd a solution S (an object that meets a particular speci cation, in this case an assignment that satis es each clause) If no such solution exists, we must say so More speci cally, a search problem must have the property that any proposed solution S to an instance I can be quickly checked for correctness What does this entail For one thing, S must at least be concise (quick to read), with length polynomially bounded by that of I This is clearly true in the case of SAT, for which S is an assignment to the variables To formalize the notion of quick checking, we will say that there is a polynomial-time algorithm that takes as input I and S and decides whether or not S is a solution of I For SAT, this is easy as it just involves checking whether the assignment speci ed by S indeed satis es every clause in I Later in this chapter it will be useful to shift our vantage point and to think of this ef cient algorithm for checking proposed solutions as de ning the search problem Thus: A search problem is speci ed by an algorithm C that takes two inputs, an instance I and a proposed solution S, and runs in time polynomial in |I| We say S is a solution to I if and only if C(I, S) = true
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Given the importance of the SAT search problem, researchers over the past 50 years have tried hard to nd ef cient ways to solve it, but without success The fastest algorithms we have are still exponential on their worst-case inputs Yet, interestingly, there are two natural variants of SAT for which we do have good algorithms If all clauses contain at most one positive literal, then the Boolean formula is called a Horn formula, and a satisfying truth assignment, if one exists, can be found by the greedy algorithm of Section 53 Alternatively, if all clauses have only two literals, then graph theory comes into play, and SAT can be solved in linear time by nding the strongly connected 231
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