Maximum satisfiability problem

In computational complexity theory, the maximum satisfiability problem (MAX-SAT) is the problem of determining the maximum number of clauses, of a given Boolean formula in conjunctive normal form, that can be made true by an assignment of truth values to the variables of the formula. It is a generalization of the Boolean satisfiability problem, which asks whether there exists a truth assignment that makes all clauses true.

The conjunctive normal form formula

( x 0 ∨ x 1 ) ∧ ( x 0 ∨ ¬ x 1 ) ∧ ( ¬ x 0 ∨ x 1 ) ∧ ( ¬ x 0 ∨ ¬ x 1 ) {\displaystyle (x_{0}\lor x_{1})\land (x_{0}\lor \lnot x_{1})\land (\lnot x_{0}\lor x_{1})\land (\lnot x_{0}\lor \lnot x_{1})}

is not satisfiable: no matter which truth values are assigned to its two variables, at least one of its four clauses will be false. However, it is possible to assign truth values in such a way as to make three out of four clauses true; indeed, every truth assignment will do this. Therefore, if this formula is given as an instance of the MAX-SAT problem, the solution to the problem is the number three.

The MAX-SAT problem is OptP-complete,^[1] and thus NP-hard (as a decision problem), since its solution easily leads to the solution of the boolean satisfiability problem, which is NP-complete.

It is also difficult to find an approximate solution of the problem, that satisfies a number of clauses within a guaranteed approximation ratio of the optimal solution. More precisely, the problem is APX-complete, and thus does not admit a polynomial-time approximation scheme unless P = NP.^[2][3][4]

More generally, one can define a weighted version of MAX-SAT as follows: given a conjunctive normal form formula with non-negative weights assigned to each clause, find truth values for its variables that maximize the combined weight of the satisfied clauses. The MAX-SAT problem is an instance of Weighted MAX-SAT where all weights are 1.^[5][6][7]

Randomly assigning each variable to be true with probability 1/2 gives an expected 2-approximation. More precisely, if each clause has at least k variables, then this yields a (1 − 2^−k)-approximation.^[8] This algorithm can be derandomized using the method of conditional probabilities.^[9]

MAX-SAT can also be expressed using an integer linear program (ILP). Fix a conjunctive normal form formula F with variables x₁, x₂, ..., x_n, and let C denote the clauses of F. For each clause c in C, let S⁺_c and S⁻_c denote the sets of variables which are not negated in c, and those that are negated in c, respectively. The variables y_x of the ILP will correspond to the variables of the formula F, whereas the variables z_c will correspond to the clauses. The ILP is as follows:

maximize	∑ c ∈ C w c ⋅ z c {\displaystyle \sum _{c\in C}w_{c}\cdot z_{c}}		(maximize the weight of the satisfied clauses)
subject to	z c ≤ ∑ x ∈ S c + y x + ∑ x ∈ S c − ( 1 − y x ) {\displaystyle z_{c}\leq \sum _{x\in S_{c}^{+}}y_{x}+\sum _{x\in S_{c}^{-}}(1-y_{x})}	for all c ∈ C {\displaystyle c\in C}	(clause is true iff it has a true, non-negated variable or a false, negated one)
	z c ∈ { 0 , 1 } {\displaystyle z_{c}\in \{0,1\}}	for all c ∈ C {\displaystyle c\in C} .	(every clause is either satisfied or not)
	y x ∈ { 0 , 1 } {\displaystyle y_{x}\in \{0,1\}}	for all x ∈ F {\displaystyle x\in F} .	(every variable is either true or false)

The above program can be relaxed to the following linear program L:

maximize	∑ c ∈ C w c ⋅ z c {\displaystyle \sum _{c\in C}w_{c}\cdot z_{c}}		(maximize the weight of the satisfied clauses)
subject to	z c ≤ ∑ x ∈ S c + y x + ∑ x ∈ S c − ( 1 − y x ) {\displaystyle z_{c}\leq \sum _{x\in S_{c}^{+}}y_{x}+\sum _{x\in S_{c}^{-}}(1-y_{x})}	for all c ∈ C {\displaystyle c\in C}	(clause is true iff it has a true, non-negated variable or a false, negated one)
	0 ≤ z c ≤ 1 {\displaystyle 0\leq z_{c}\leq 1}	for all c ∈ C {\displaystyle c\in C} .
	0 ≤ y x ≤ 1 {\displaystyle 0\leq y_{x}\leq 1}	for all x ∈ F {\displaystyle x\in F} .

The following algorithm using that relaxation is an expected (1-1/e)-approximation:^[10]

1. Solve the linear program L and obtain a solution O
2. Set variable x to be true with probability y_x where y_x is the value given in O.

This algorithm can also be derandomized using the method of conditional probabilities.

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:

1. Run the (derandomized) 1/2-approximation algorithm to get a truth assignment X.
2. Run the (derandomized) (1-1/e)-approximation to get a truth assignment Y.
3. Output whichever of X or Y maximizes the weight of the satisfied clauses.

This is a deterministic factor (3/4)-approximation.^[11]

On the formula

F = ( x ∨ y ) ⏟ weight  1 ∧ ( x ∨ ¬ y ) ⏟ weight  1 ∧ ( ¬ x ∨ z ) ⏟ weight  2 + ϵ {\displaystyle F=\underbrace {(x\lor y)} _{{\text{weight }}1}\land \underbrace {(x\lor \lnot y)} _{{\text{weight }}1}\land \underbrace {(\lnot x\lor z)} _{{\text{weight }}2+\epsilon }}

where ϵ > 0 {\displaystyle \epsilon >0} , the (1-1/e)-approximation will set each variable to True with probability 1/2, and so will behave identically to the 1/2-approximation. Assuming that the assignment of x is chosen first during derandomization, the derandomized algorithms will pick a solution with total weight 3 + ϵ {\displaystyle 3+\epsilon } , whereas the optimal solution has weight 4 + ϵ {\displaystyle 4+\epsilon } .^[12]

The state-of-the-art algorithm is due to Avidor, Berkovitch and Zwick,^[13][14] and its approximation ratio is 0.7968. They also give another algorithm whose approximation ratio is conjectured to be 0.8353.

Many exact solvers for MAX-SAT have been developed during recent years, and many of them were presented in the well-known conference on the boolean satisfiability problem and related problems, the SAT Conference. In 2006 the SAT Conference hosted the first MAX-SAT evaluation comparing performance of practical solvers for MAX-SAT, as it has done in the past for the pseudo-boolean satisfiability problem and the quantified boolean formula problem. Because of its NP-hardness, large-size MAX-SAT instances cannot in general be solved exactly, and one must often resort to approximation algorithms and heuristics^[15]

There are several solvers submitted to the last Max-SAT Evaluations:

• Branch and Bound based: Clone, MaxSatz (based on Satz), IncMaxSatz, IUT_MaxSatz, WBO, GIDSHSat.
• Satisfiability based: SAT4J, QMaxSat.
• Unsatisfiability based: msuncore, WPM1, PM2.

MAX-SAT is one of the optimization extensions of the boolean satisfiability problem, which is the problem of determining whether the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to TRUE. If the clauses are restricted to have at most 2 literals, as in 2-satisfiability, we get the MAX-2SAT problem. If they are restricted to at most 3 literals per clause, as in 3-satisfiability, we get the MAX-3SAT problem.

There are many problems related to the satisfiability of conjunctive normal form Boolean formulas.

• Decision problems:
  • 2SAT
  • 3SAT
• Optimization problems, where the goal is to maximize the number of clauses satisfied:
  • MAX-SAT, and the corresponded weighted version Weighted MAX-SAT
  • MAX-kSAT, where each clause has exactly k variables:
    • MAX-2SAT
    • MAX-3SAT
    • MAXEkSAT
  • The partial maximum satisfiability problem (PMAX-SAT) asks for the maximum number of clauses which can be satisfied by any assignment of a given subset of clauses. The rest of the clauses must be satisfied.
  • The soft satisfiability problem (soft-SAT), given a set of SAT problems, asks for the maximum number of those problems which can be satisfied by any assignment.^[16]
  • The minimum satisfiability problem.
• The MAX-SAT problem can be extended to the case where the variables of the constraint satisfaction problem belong to the set of reals. The problem amounts to finding the smallest q such that the q-relaxed intersection of the constraints is not empty.^[17]

• Boolean Satisfiability Problem
• Constraint satisfaction
• Satisfiability modulo theories

• http://www.satisfiability.org/
• https://web.archive.org/web/20060324162911/http://www.iiia.csic.es/~maxsat06/
• http://www.maxsat.udl.cat
• Weighted Max-2-SAT Benchmarks with Hidden Optimum Solutions
• Lecture Notes on MAX-SAT Approximation

• Vazirani, Vijay V. (2001), Approximation Algorithms (PDF), Springer-Verlag, ISBN 978-3-540-65367-7