# MaxSAT Solving

Given an unsatisfiable formula in CNF, the MaxSAT problem is the problem of finding an assignment which satisfies the maximum number of clauses and therefore solves an optimization problem rather than a decision problem. There are two extensions to MaxSAT Solving: 1) the distinction of hard/soft clauses, and 2) additional weighted clauses, yielding four different flavours of MaxSAT solving:

1. Pure MaxSAT
2. Partial MaxSAT
3. Weighted MaxSAT
4. Weighted Partial MaxSAT

In a Partial MaxSAT problem you can distinguish between hard and soft clauses. A hard clause must be satisfied whereas a soft clause should be satisfied but can be left unsatisfied. This means the solver only optimizes over the soft clauses. If the hard clauses themselves are unsatisfiable, no solution can be found.

In a Weighted MaxSAT problem clauses can have a positive weight. The solver then does not optimize the number of satisfied clauses but the sum of the weights of the satisfied clauses.

The Weighted Partial MaxSAT problem is the combination of Partial MaxSAT and weighted MaxSAT. I.e. you can add hard clauses and weighted soft clauses to the MaxSAT solver.

Note two important points:

1. MaxSAT can be defined as weighted MaxSAT restricted to formulas whose clauses have weight 1, and as Partial MaxSAT in the case that all the clauses are declared to be soft.
2. The above definitions talk about clauses on the solver, not arbitrary formulas. In real-world use cases you often want to weight whole formulas and not just clauses. LogicNG's MaxSAT solver API gives you this possibility and internally translates the formulas and their weights accordingly. This is a huge quality-of-life improvement when working with the solvers.

## MaxSAT Algorithms in LogicNG

LogicNG's MaxSAT implementation is based on OpenWBO and its underlying encodings with PBLib. Therefore, you can use different algorithms, depending on your MaxSAT flavour and your specific use case. There are two orthogonal strategies to solve MaxSAT problems:

1. Based on linear search
2. Based on producing and iteratively relaxing unsatisfiable cores

Details on this distinction can be found in the paper Algorithms for Weighted Boolean Optimization by Manquinho, Marques-Silva and Planes.

Here is an overview of the algorithms in LogicNG and their capabilities:

Algorithm Description Strategy Unweighted Weighted
LinearSU Linear Sat-Unsat Linear Search yes yes
LinearUS Linear Unsat-Sat Linear Search yes no
MSU3 Seminal-core guided algorithm Unsat-Core yes no
WMSU3 Weighted MSU3 Unsat-Core no yes
WBO Weighted Boolean Optimization Unsat-Core yes yes
IncWBO Incremental WBO Unsat-Core yes yes
OLL OLL (since 2.4.0) Unsat-Core yes yes

All algorithms in LogicNG support partial MaxSAT. LinearUS and MSU3 do not support weighted clauses/formulas, WMSU3 does not support unweighted clauses/formulas.

The LinearSU stands for Linear SAT-UNSAT and it means that all sat-calls on the SAT-Solver are SAT except for the last call. That means: We start with a version without soft formulas on the SAT solver (which is SAT as long as the conjunction of the hard formulas evaluate to true). Then we add clauses as long as an UNSAT comes for the first time. LinearUS stands for Linear UNSAT-SAT and works the other way around: All SAT-Solver calls are UNSAT and the last one is SAT. So first we add all soft clauses, then remove clauses one by one until the SAT-call is SAT.

The three MSU3, WMSU3, and OLL algorithms are based on unsatisfiable cores. The two WBO and IncWBO algorithms are based on a CNF encoding of the pseudo-Boolean Optimization problem and therefore are strongly dependent on the size of the weights.

Which algorithm to use depends strongly on the specific use case and must be evaluated. In practice OLL should be the best choice for all weighted problems especially when dealing with large weights.

## Using MaxSAT Solvers in LogicNG

A MaxSAT solver in LogicNG is created by using its corresponding factory method in the class MaxSATSolver. E.g. the method incWBO(f) generates a MaxSAT solver with the IncWBO algorithm. Each factory method can have an optional parameter for the solver's configuration.

Hard formulas are added with the method addHardFormula(), soft clauses are added with addSoftClause(formula, weight) where the weight has to be positive. For an unweighted clause, use the weight 1. The method solve() solves the problem for the formulas on the solver and returns a MaxSATResult which has one of three states:

1. UNSATISFIABLE: The hard clauses on the solver are already unsatisfiable, optimization could not be performed
2. OPTIMUM: An optimal solution was found
3. UNDEF: The search was aborted by a handler and yielded no result

As with SAT solvers, a MaxSAT solve process can be customized with a handler which can be implemented by the user and aborts the solve process in certain conditions.

The method model() returns the model found by the solver when an optimal solution was found. The method result() returns the minimum weight (or number of clauses if unweighted) of clauses/formulas which have to be unsatisfied. Therefore, if the minimum number of weights is 0, the formula is satisfiable.

We look at some examples on how to use the MaxSAT solver with the following simple set of formulas which are unsatisfiable together:

A & B & (C | D)
A => ~B
~C
~D


### Pure MaxSAT Solving

We generate a MSU3 solver and add all formulas as soft clauses with weight 1. Therefore, we have a pure MaxSAT problem without distinction between hard/soft clauses and without weights.

MaxSATSolver solver = MaxSATSolver.msu3(f);
solver.addSoftFormula(f.parse("A & B & (C | D)"), 1);

solver.solve();  // OPTIMUM
solver.model();  // Assignment{pos=[], neg=[~A, ~B, ~C, ~D]}
solver.result(); // 1


In this case we can find an optimal solution: at least one formula remains unsatisfied. In this case the model assigns all variables to false, thus the first formula is unsatisfied and all other formulas are satisfied.

As we will see in the next section, this is the only way of unsatisfying only one formula. If the first formula holds, we have to unsatisfy at least two other formulas.

### Partial Weighted MaxSAT Solving

We generate a WMSU3 solver and add once again the first formula as hard clause and all other formulas as soft clauses with different weights. Therefore, we have a partial weighted MaxSAT problem.

MaxSATSolver solver = MaxSATSolver.wmsu3(f);
solver.addHardFormula(f.parse("A & B & (C | D)"));

solver.solve();  // OPTIMUM
solver.model();  // Assignment{pos=[A, B, C], neg=[~D]}
solver.result(); // 6


In this case we can find an optimal solution once again. Since the fourth formula now has a large weight, the incentive to satisfy it is larger than the incentive to satisfy the third formula. So in contrast to the last section the model now includes C and not D, thus unsatisfying the third formula.

The result in this case is 6 - the sum of weights of all unsatisfied formulas: the second formula with weight 2 and the third formula with weight 4.

Multiple Optimal Solutions

Note that this is the only optimal solution for this problem. In general, however, there can be multiple optimal solutions.

### Special Cases

Last we consider two special cases. First we try to solve a formula where the hard clauses are already unsatisfiable:

MaxSATSolver solver = MaxSATSolver.msu3(f);

solver.solve();  // UNSATISFIABLE
solver.model();  // null
solver.result(); // -1


In this case the solve() method returns the state UNSATISFIABLE. There is no model and the result is -1.

Now we try a formula which is satisfiable:

MaxSATSolver solver = MaxSATSolver.msu3(f);
MaxSATSolver solver = MaxSATSolver.msu3(f);

solver.solve();  // OPTIMUM
solver.model();  // Assignment{pos=[A, B, C, D, X], neg=[~Y]}
solver.result(); // 0


The solver found an optimal solution in this case. Further, the result is 0 - indicating that all formulas were satisfied and no formula had to be unsatisfied.

## Minimum vs. Maximum: Solving the Dual Problem

As seen above, the solver tries to maximize the number of satisfied clauses by minimizing the number of unsatisfied clauses. The returned result is the sum of weights of the unsatisfied clauses.

But what when you want to know the minimal number of satisfied clauses? Then you have to adjust your problem modelling accordingly. Let us consider a small example:

MaxSATSolver solver = MaxSATSolver.wmsu3(f);
solver.addHardFormula(f.parse("(A & B & ~X & ~D) | (B & C & ~A) | (C & ~D & X)"));


We have one hard formula and each variable in it has a given weight. Solving the problem yields the assignment Assignment{pos=[B, C, D, X], neg=[~A]} with result 2. This means when setting all variables except A to true, only one formula (in this case variable) is unsatisfied and has weight 2.

If we give the variable weights a meaning, e.g. a price, then we have now computed the most expensive variable assignment. B, C, D, X together have a price of 24. In order to compute the cheapest variable assignment, we should not try to maximize the positive literals, but instead the negative literals: We want to set a maximal number of expensive variables to false. We can do this by negating the variables in the soft formulas:

MaxSATSolver solver = MaxSATSolver.wmsu3(f);
solver.addHardFormula(f.parse("(A & B & ~X & ~D) | (B & C & ~A) | (C & ~D & X)"));

Now the model is Assignment{pos=[A, B], neg=[~C, ~D, ~X]} with a result of 6. This means the cheapest variable assignment consists of only A and B with a price of 6.