_{1}

This paper analyzes the resolution complexity of a random CSP model named model RB
^{mix}, the instance of which is composed by constraints with different length. For model RB
^{mix}, the existence of phase transitions has been established and the threshold points have been located exactly. By encoding the random instances into CNF formulas, it is proved that almost all instances of model RB
^{mix} have no tree-like resolution proofs of less than exponential size. Thus the model RB
^{mix} can generate abundant hard instances in the threshold. This result is of great significance for algorithm testing and complexity analysis in NP-complete problems.

Constraint satisfaction problem (CSP) is originated from artificial intelligence. It is a very important branch in the field of artificial intelligence, and it is a problem widely studied in computer science, information science, discrete mathematics and other interdisciplines. At present, a number of problems restricting the development of computer science, automatic control, system engineering and other disciplines can be modeled as CSPs. At the same time, CSPs are widely used in related application fields such as resource allocation, pattern recognition, temporal reasoning and image recognition.

CSP is composed of a set of variables and a set of constraints. Each variable takes a value from the corresponding non-empty domain. The number of elements in the domain can be fixed, or change with the number of variables. Each constraint has a corresponding incompatible assignment set to restrict the values of variables appearing in the constraint. The constraint set randomly selected constitutes a random instance of CSP. Given a random instance, if there exists an assignment to the variables satisfying all the constraints in the instance simultaneously, we call this assignment a solution. Usually, most of the CSPs are NP-complete problems. A lot of theoretical research and experimental results show that for many NP-complete problems we can define a control parameter (such as constraint density, constraint tightness, etc.). There is a critical value of the control parameter, below which the probability of an instance being satisfiable goes to 1, and above which it goes to 0 as the number of variables approaches infinity. People call this phenomenon the phase transition. The critical value is a satisfiability phase transition point.

Random k-SAT (k-satisfiability. Every clause of random k-SAT is randomly and independently generated. Each clause contains just k different variables) problem is a typical CSP with a fixed domain (The variable is Boolean, i.e. 0 and 1 or T and F), and the first NP-complete problem proved by Cook. Research shows that when

Model RB (Revised B) is a nontrivial random CSP with growing variable domain. Specifically, in order to overcome the trivial gradual unsolvability of model B in the standard model CSP [

Inspired by model ^{mix}. Interestingly, the phase transition behavior of model ^{mix} has nothing to do with the number of constraints with different lengths. It has been proved that model RB^{mix} mixed with constraints with different lengths has the exact satisfiability phase transition phenomenon similar to that of model RB, and the phase transition points can be located exactly. We know that if the new model CSP has the exact satisfiability phase transition phenomenon and can generate a large number of hard instances, the model has important significance for testing of CSP algorithm. For model RB^{mix}, we use the resolution method (its general form is for logic formula) to encode random instances generated by model RB^{mix} into the conjunctive normal form in SAT problem, i.e. CNF formula. By using five lemmas, we prove a famous theorem about the resolution length. Therefore, it is proved that the random instances generated by model RB^{mix} almost have no resolution complexity for which the resolution length is less than the exponential size. This shows that model RB^{mix} is hard for algorithms based on resolution, so model RB^{mix} can generate a large number of hard instances in the phase transition region. It is proved in this paper that almost all instances of model RB^{mix} have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of random CSP instances hard to solve, which can be useful in the experimental evaluation of CSP algorithms, but also propose a CSP model with both many hard instances and exact phase transitions.

The random instance of model RB^{mix} is composed of the variable set

generated in the following way.

Step 1:

Step 2: For each

If random instances generated by model RB^{mix} have a group of assignments of n variables, so that all

Thus, model RB^{mix} is a promotion of model RB on the constraints. In model RB, all constraints are k-ary constraints, which means that the constraint length is fixed. Model RB^{mix} is composed of constraints with different lengths. Apparently, model RB is a special circumstance of model RB^{mix} when^{mix} is more generally representative.

Assume that ^{mix}. Let

Theorem 1 Assume that

Theorem 2 Assume that

From the above two theorems, we can know that under the condition that the domain of variables is not too small (^{mix} being satisfiable suddenly changes from 1 to 0, which means that model RB^{mix} has the satisfiability phase transition phenomenon, and it can get the accurate phase transition point. Therefore, model RB^{mix} is a mixed random CSP with accurate phase transition and a large domain.

Assume that ^{mix}. We can get the following theorem.

Theorem 3 If

As we know, for the unsatisfiable random instance I, there must be the shortest resolution proof which can reason out empty clauses. Its length is the lower bound of time consumed by any algorithm based on the resolution principle. Therefore, from Theorem 3 we can know that the solution algorithm of unsatisfiable instances of model RB^{mix} has the complexity of exponential size. Thus in the phase transition region model RB^{mix} can generate a large number of hard instances.

This paper uses the resolution method to analyze the complexity of model RB^{mix}. The general form is CNF formula for SAT problem. The so-called CNF formula refers to the conjunction expression (^{mix} according to a certain rules into CNF formula, then the resolution complexity of

In model RB^{mix}, assume that the domain of each variable

(1) Clauses of the domain： Each variable

(2) A value clause at most: Ensure that each variable takes a value from its domain at most at a time, i.e.

(3) Conflicted clause: It is used to remove the assignment in the uncoordinated set, so that the constraint is satisfied. For example,

Definition 1 (Length of the clause) The number of variables appearing in Clause

Definition 2 (Length of CNF) The maximum length of all the clauses in F of

CNF formula F is called the length of F, i.e.

Definition 3 (Resolution) The clause sequence

Definition 4 (Derived length) The length for

Definition 5 (Resolution) The resolution in which an empty clause (expressed with

We consider the clause in

For the unsatisfiable formula F, there must be the shortest resolution proof deriving

Ben-Sasson and Wigderson gave the following theorem used to prove the lower bound of the resolution length of CNF formula [

Theorem 4 Assume that F is a CNF formula with n variables, then we have:

We want to use Theorem 4 to prove the important conclusion Theorem 3 in this paper. We need to use the following five lemmas.

Before giving Lemma 1, we first give the following definition.

Definition 6 (Subproblem) Assume that I is an instance of a CSP, then the instance composed of subsets of constraint set in I is called a subproblem of I.

Lemma 1 Assume that I is a random instane generated by model RB^{mix}, then

Proof Assume that A = {I has at least

Let

For the sufficiently large n, here exists a constant

Let

thus

Therefore,

Before giving Lemma 2, we first give the following definitions.

Definition 7 (i-constraint assignment group) Assume that i constraints contain variable x. If assignments are made for variables except x in i constraints, the such assignment group is called i-constraint assignment group, written as

Definition 8 (flawed i-constraint assignment) The assignment of given variable x is

Lemma 2 Assume that I is a random instance generated by model RB^{mix}, then for

Proof Assume that A = {I has a defective i-constraint assignment group

Next, let B = {

^{mix}，

Here,

because

And for

From

Thus,

Because the maximum choices of i-constraint assignment group

And

Thus, we have

So

Lemma 3 Assume that I is a random instance generated by model RB^{mix}, then almost any subproblem with the size of

Proof Assume that I has the unsatisfiable subproblem with the size of

Before giving Lemma 4, we need to give the following definition.

Definition 9 (Complexity) Assume that I is a random CSP instance. Encode I into CNF formula F. Assume that F’s clause C is derived from sequence

Lemma 4 Assume that I is a random instance generated by model RB^{mix}. Encode I into CNF formula F. Then almost any inversion of F has a clause which satisfies

Proof From Lemma 3 we can know that

Lemma 5

Proof Assume that I is a random instance generated by model RB^{mix}. Encode I into CNF formula F. Assume that

Assume that variable x whose degree is not greater than not

So

Finally, from Lemma 5 and Theorem 4, we can get Theorem 3. That is unsatisfiable random instances generated by model RB^{mix }have no resolution proof whose complexity is less than the exponential size.

In this paper, we analyze the resolution complexity of a mixed constraint satisfaction problem named model RB^{mix}. It is theoretically proved that unsatisfiable random instances generated by model RB^{mix }almost have no complexity whose resolution length is smaller than the exponential size. As a result, based on the resolution algorithm the random instance of model RB^{mix} has complexity of exponential size. The conclusion shows that model RB^{mix} with different length constraints not only has accurate satisfiability phase transition phenomenon, but also can generate a large number of hard instances in the phase transition region. Therefore, model RB^{mix} is a large-domain nontrivial random constraint satisfaction problem with mixed constraints and accurate phase transition. In the later work, we can also verify the hardness of model RB^{mix} in the experiment, and further study the solving algorithm.

Shi, X.M. (2017) Complexity Analysis of Mixed Constraint Satisfaction Problems. Open Journal of Ap- plied Sciences, 7, 129-139. https://doi.org/10.4236/ojapps.2017.74011