Problem Formulation

4.3.1Nonlinear Integer Programming Formulation

We now formulate CSAP as a nonlinear integer programming (NLIP) problem.

To reduce broadcasting delay and achieve fairness on resource sharing, we use two metrics in our problem: broadcasting delay (denoted by D) and fairness index (denoted by I). Since we assume that the capacity of each channel is the same and all SUs are always backlogged with packets, it is reasonable to use Jain’s fairness index as our fairness metric. If there are m SUs and SU i can obtain resource units, then Jain’s fairness index is given by:

　　　(38)

The larger the fairness index, the fairer the resource allocation. The maximum value of Jain’s fairness index is 1. Since the two metrics, I and D, are in different units,

we use the ratio of fairness index and broadcasting delay (called ID ratio) as our metric of channel selection strategy. With a larger fairness index and a fixed broadcasting delay, the ID ratio is increased. Similarly, with a smaller broadcasting delay and a fixed fairness index, the ID ratio is also increased.

Hence, the objective of this formulation is to maximize the ID ratio. Given an SBS with available channel set and m SUs with available channel sets,

, we can obtain the common available channel information (denoted by C) between the SBS and each SU. Let C= , where i = 0, …, n and j = 1, …, m.

Each indicator variable is defined as follows:

　　　(39)

With C, we can formulate CSAP via NLIP. We first define two decision variables:

Then the objective function and the constraints of this channel selection and assignment problem are described as:

(40)

(41)

(42)

(43)

(44)

Since there is the same amount of resource in each data sub-frame left for unicasting service, we view the resource as one resource unit. Since 1) the portion of a resource

unit each SU can obtain for unicasting service on a channel is determined by the number of SUs to which the channel is assigned, and 2) the SUs to which a channel is assigned will equally share the resource unit, (38) can be transferred to I in (40). The constraint in (41) means that for each SU, at least one channel of all common available channels between the SBS and the SU should be selected. In other words, the broadcasting packets should be sent to all SUs which are being associated with the SBS. The constraint in (42) states that if a channel is selected, the channel should be assigned to at least one SU. When a channel is selected and is not assigned to any SU, then there is no need to send broadcasting packets on the channel. The constraint in (43) indicates that if a channel is not selected, the channel cannot be assigned to any SU. The constraint in (44) means that exactly one channel can be assigned to each SU.

In what follows, we will show that CSAP is NP-hard. Then we propose an enumerative algorithm to find an optimal solution and design a greed-based heuristic algorithm with polynomial-time complexity.

4.3.2 Problem Complexity Analysis

We will prove that CSAP is NP-hard through applying a reduction from a well-known NP-complete problem called 3-dimensional matching problem. From [30], the 3-dimensional (3D) matching problem is defined as follows.

Definition 4.1. Given three disjoints sets , ,

and a family of triples with

for . The 3D matching problem is to determine whether there exists an which is a subset of F, , and .

Theorem 4.2. The channel selection and assignment problem CSAP is NP-hard.

Proof. It is clear that when n is smaller than m, the answer to the 3D matching

problem is no. Hence, we assume that n is larger than m in this reduction.

Given an instance of the 3D matching problem with three disjoint sets

, , and a family , we can

construct an instance of CSAP with n channels, 3m SUs, an SBS and common available channel information, for i = 1, …, n and for all . The n channels correspond to the elements of and the 3m SUs correspond to the elements of . Each indicator variable is defined as follows.

It is clear that the reduction can be completed in polynomial time. In addition, it can be shown that there is a channel selection and assignment strategy with optimal ID ratio of if and only if there is a 3D matching denoted by . If there is a 3D matching, the channels corresponding to the elements of are selected. For each of , Channel is assigned to SU , SU , and SU . According to (40), D and I of this channel selection and assignment strategy are m and 1, respectively. The reason is that m channels are selected and each channel is assigned to only three SUs. Since the SBS can connect only three different SUs through one channel, at least m channels will be selected. Then the minimal value of D is m. By definition of Jain’s fairness index, the maximal value of I is 1. Hence, the channel selection and assignment strategy has an optimal ID ratio of . If there is a channel selection and assignment strategy with an optimal ID ratio of , it is clear that at most m channels are selected. According to the constraint in (44) and the SBS only connecting three different SUs through one channel, the number of channels selected is exactly m. Hence, it is clear that the set of triples corresponding to those m channels is 3D matching. ■

Algorithm 1

Input: M, C and all possible channel assignment strategies.

Output: S.

1: S = 2: MAX = 0

3: for each possible channel assignment strategy, S’ do 6: for j = 1 to |M| do

7:

8:

10: end for