An important goal for studying genetic regulatory network is to understand the gene behavior and to develop optimal control policy for potential applications to medical therapy. While many models have been proposed for modeling gene regulatory networks, Boolean Networks (BNs) [1–3] and thier extension Probabilistic Boolean Networks (PBNs) [4] have received much attention. Because they form a class of models which can capture the logical interactions of genes and they are also effective in modeling pathways for drug discovery [5]. Recently applications in medical treatment for Parkinson's disease can also be found in [6]. In fact, a PBN can be considered as a collection of BNs driven by a Markov chain and therefore its dynamics and behavior can be studied by using Markov chain theory. For reviews on BNs and PBNs, we refer interested readers to [7–9] and the references therein.

Many methods in control theory are available for the intervention of PBNs. A gene control model has been proposed in [10]. The control model is formulated as a mixed integer programming problem and it aims at driving the PBN from the undesirable states to the desirable ones. A class of PBN control problems with hard constraints has been proposed in [11, 12]. The motivation of the control model is to reduce the side-effects of medical treatment. In [11], hard constraints are included in the optimal control problem and an approximation method is then proposed in [12] to obtain the optimal controls efficiently.

Datta et al. [13] proposed an external intervention method based on optimal control theory. In their work, genes are classified as internal nodes and external nodes (control nodes). One can intervene the values of internal nodes in some desirable manner by controlling the values of certain external nodes. By defining the control cost for each control input and terminal cost for each state, the problem is to find a sequence of control inputs that leads the network into desirable states at the terminal step with minimum average cost. The classical technique of dynamic programming is then employed to solve the optimal control problem.

Chen et al. [14] then consider an external intervention problem based on optimal control theory and dynamic programming. Given the terminal cost of each state, the objective is to drive the network into the state with the maximum cost being minimized by applying external controls. The problem is important in the view of medical therapy because patients/organisms would like to minimize the damage even for the worst case. They proved that both minimizing the maximum cost and minimizing the average cost are
. A dynamic programming-based algorithm is then proposed for finding a control sequence that minimizes the maximum cost in control of PBN. The above dynamic programming-based methods still have high computational complexity. The size of the underlying transition probability matrix increases exponentially with the number of nodes in the PBN. To tackle this problem a possible remedy is to consider network reduction approach.

Several reduction methods have been proposed recently. In [15], a CoD-based reduction algorithm is introduced. Coefficient of Determination (CoD) helps to evaluate the influence of a candidate node for deletion on the target node and find the optimal candidate node for deletion. The proposed algorithm can well preserve the attractor structure and long-run dynamics of the original network.

Qian et al. [16] proposed a state reduction method by considering deleting states directly. Instead of deleting the nodes in a network, they delete the out-most states having less influence to the network. Here we consider a transition probability-based reduction strategy. This strategy is easy to implement as we do not need to compute the stationary distribution of the PBN beforehand.

We consider the problem of minimizing the maximum cost in control of PBN and we employ transition probability-based reduction strategy to reduce the network complexity of a PBN. We show that under some condition and in many of our numerical examples, the optimal control sequence obtained from the reduced network is the same as the one in the original network. Then we apply the dynamic programming-based algorithm to the reduced network. The computational complexity of dynamic programming-based algorithm when applied to the original network is *O*(2^{
n
}) (depending on the number of network states) when the number of control nodes *m* and the number of steps *M* are fixed. When our state reduction method is applied, the computational complexity is reduced to *O*(*|R|*), where *R* is the set of states after reduction.

The remainder of the paper is structured ae follows. We first give a brief review on PBNs and the dynamic programming method. We then introduce our state reduction approach together with some theoretical results to support our proposed approach. Numerical examples are given to demonstrate both the effectiveness and the efficiency of our proposed method. Finally some discussion will be given to conclude the paper.

### A brief review on BNs and PBNs

A BN consists of a set of *n* nodes (genes) as follows: {*v*
_{1},*v*
_{2},..., *v*
_{
n
}}, *v*
_{
i
}∈ {0,1} and a set of Boolean functions denoted by {*f*
_{1}, *f*
_{2},..., *f*
_{
n
}}. Each *v*
_{
i
}(*t*) is defined as the state of node *i* at time *t*. The rules of regulatory interactions among nodes are then represented by the Boolean functions: *v*
_{
i
}(*t +* 1) *= f*
_{
i
}(*v*
_{
i
}
_{1}, *v*
_{
i
}
_{2},..., *v*
_{
ik
}) where {*v*
_{
i
}
_{1},*v*
_{
i
}
_{2},...,*v*
_{
ik
}} are input nodes of *f*
_{
i
}, and they are called parent nodes of node *v*
_{
i
}. We define *IN*(*vi*) = {*v*
_{
i
}
_{1}, *v*
_{
i
}
_{2},..., *v*
_{
ik
}}. The number of parent nodes to *v*
_{
i
}is called the *in-degree* of *v*
_{
i
}. The largest in-degree of {*v*
_{1}, *v*
_{2},..., *v*
_{
n
}} is called the *maximum in-degree* of BN and is denoted by *K*.

Since BN is a deterministic model, a stochastic model is more preferable due to the measurement noise in inferring a gene regulatory network. A stochastic version of BN, PBN [

4,

9] is then introduced to cope with the weakness. A PBN can be regarded as an extension of BN to a probabilistic setting. In a PBN, each node

*v*
_{
i
}has a set of Boolean functions:

The state of

*v*
_{
i
}at time

*t +* 1 is predicted by one of the Boolean functions in (1) with selection probabilities

. Here

A PBN can be regarded as a finite collection of BNs over a fixed set of nodes, where each BN has a fixed set of Boolean functions

. The BN having Boolean function set

**f**
_{
j
}(

*j =* 1,2,...,

*N*) is called the

*j*th BN. At each time step

*t*, the selection process of Boolean functions is assumed to be independent, and the selection probability is given by

and the states of {

*v*
_{1}(

*t +* 1),

*v*
_{2}(

*t* + 1),...,

*v*
_{
n
}(

*t +* 1)} is predicted by the Boolean function set

**f**
_{
j
}. Then we introduce the decimal representation of states. Suppose the current state is {

*v*
_{1}(

*t*),

*v*
_{2}(

*t*),...,

*v*
_{
n
}(

*t*)}, we define

Since *v*
_{
i
}(*t*) ∈ {0,1}, *w*(*t*) can take any integral value in [1, 2^{
n
}].

The dynamics of a PBN can be studied by using Markov chain theory, see for instance [

17]. The one-step transition probability can be represented by using the transition probability matrix

*A* where each entry

*A*
_{
ij
} is given by

Here *i = w*(*t +* 1) and *j = w*(*t*) and
is set of BNs that the network can enter state *i* from state *j*. We remark that *A* is a column stochastic matrix, i.e.,
.