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In the maximum-2-satisfiability problem (MAX-2-SAT), the input is a formula in conjunctive normal form with two literals per clause, and the task is to determine the maximum number of clauses that can be simultaneously satisfied by an assignment. Like the more general maximum satisfiability problem, MAX-2-SAT is NP-hard.
The 1/2-approximation algorithm does better when clauses are large whereas the (1-1/e)-approximation does better when clauses are small. They can be combined as follows: Run the (derandomized) 1/2-approximation algorithm to get a truth assignment X. Run the (derandomized) (1-1/e)-approximation to get a truth assignment Y.
The exponential time hypothesis asserts that no algorithm can solve 3-SAT (or indeed k-SAT for any k > 2) in exp(o(n)) time (that is, fundamentally faster than exponential in n). Selman, Mitchell, and Levesque (1996) give empirical data on the difficulty of randomly generated 3-SAT formulas, depending on their size parameters.
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are ...
In computer science and formal methods, a SAT solver is a computer program which aims to solve the Boolean satisfiability problem (SAT). On input a formula over Boolean variables, such as "(x or y) and (x or not y)", a SAT solver outputs whether the formula is satisfiable, meaning that there are possible values of x and y which make the formula true, or unsatisfiable, meaning that there are no ...
For every ε > 0, there is a PCP-verifier M for 3-SAT that reads a random string r of length ( ()) and computes query positions i r, j r, k r in the proof π and a bit b r. It accepts if and only if 'π(i r) ⊕ π(j r) ⊕ π(k r) = b r. The Verifier has completeness (1−ε) and soundness 1/2 + ε (refer to PCP (complexity)). The ...
There is a simple randomized polynomial-time algorithm that provides a ()-approximation to MAXEkSAT: independently set each variable to true with probability 1 / 2 , otherwise set it to false. Any given clause c is unsatisfied only if all of its k constituent literals evaluates to false.
Both algorithms work on formulae in Boolean logic that are in, or have been converted into conjunctive normal form. They start by assigning a random value to each variable in the formula. If the assignment satisfies all clauses, the algorithm terminates, returning the assignment. Otherwise, a variable is flipped and the above is then repeated ...