is a pair (
(k is the length of the constraint) is a subset of U, and
is a compatible assignments set.
is satisfied if the k-tuple of values assigned to variables in
is contained in
. A solution of a CSP instance is an assignment to all the variables that satisfies all constraints.
2.2. d-p-RB Model
A random CSP instance in d-p-RB model is generated in the following two steps:
Step 1. We select with repetition l groups of constraints. For each group, there are
constraints with each contains k variables, which are randomly select from u, and distinct from each other.
Step 2. For each group of constraints, we uniformly select at random without repetition
is the constraint tightness) compatible assignments to form the compatible assignments set
3. Main Result
, where r is a constant control parameter, which determines how many constraints are in a CSP instance. Let
) determines how restrictive the constraints are. Let
denote the probability of a random d-p-RB instance being satisfiable, then we have the following theorem.
), if the constants k, p, α, γ satisfy the relations
The theorem shows that, when the number of variables n is sufficiently large, there exists a sudden shift in
4. Proof of the Theorem
Let N denote the number of solutions of a random CSP instance I. The expectation and the second moment of N is denoted by
, we consider the Markov inequality
the second moment method, we estimate the upper bound of
, and then by the Cauchy inequality
, we finally attain our goal. Now we demonstrate the two cases respectively.
4.1. Proof of r > rm
Since the constraints are generated independently in d-p-RB model, the expected number of solutions
is given by
, we have
Then using the Markov inequality
, by (5), it’s not hard to have
4.2. Proof of r < rm
Definition 1 (The assignment pair) Suppose that the assignment pair
is an ordered pair, where
). An assignment pair
satisfies a CSP instance if and only if both
satisfy the instance.
Definition 2 (The similarity number) Define a function as follows
, thus the assignment pair
identical assignments, i.e. the similarity number of
is m. It is obvious that
Next, we use the second moment method to complete the proof.
represents the probability that
satisfy the instant I simultaneously. We analyse this probability in the following way:
Since there are m identical assignments in
, for each constraint, we have the following two cases:
1) The assignments of k variables that the constraint restricts are all same in
, in this case, the probability of
satisfying the constraint is
, and for a random constraint, the probability of such a situation is
2) Otherwise, the probability of
satisfying the constraint is
, and the probability that
falls into such a situation is
Since the constraints are generated independently, the assignment pair
satisfying all the constraints in random instance i is
be the set of assignment pairs whose similarity number is m,
be the cardinality of
, then we have
Thus by (10) and (11), the second order moment of the number of solutions of the random instance of d-p-RB model is
, in order to evaluate the upper bound of the above Inequality (15), we divide the interval [0,1] into three parts:
, here β and γ satisfy
, recalling that
, we get
Then it is not hard to obtain that
, by the Stirling’s formula
, it’s not hard to see
, we have
, we have
Thus, for arbitrary small ε, there exists an integer
, such that
with respect to s, we get
By the condition
, we have
, which implies that
is convex for
. Note that
, therefore we get
is a constant.
, similarly we have
is concave and has the maximum value
. Thus we have
So we get
, there exists an integer
, such that
Summarizing the above, from (18), (25), (35), letting
, we obtain
Thus we have
then by the Cauchy inequality
, we have
then we get
Thus the theorem is proved.
So far we have demonstrated the satisfiability phase transition in theory. From the proof of the theorem, it can be seen that when the control parameter r is less than the transition point
, the probability of a CSP instance being satisfied tends to 1, while the control parameter r is greater than the transition point
, the probability tends to 0. Thus there exists a sharp threshold in the CSP instances generated by d-p-RB model.
In this paper, we propose a new CSP model d-p-RB. Compared with RB model, we diversify the constraint tightness p and broaden the domain size d. By the method of second moment, we proved that there indeed exist satisfiability phase transition phenomenon and the transition point can also be located exactly.
Cite this paper
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