- Research Article
- Open Access

# On Coverage and Capacity for Disaster Area Wireless Networks Using Mobile Relays

- Wenxuan Guo
^{1}Email author and - Xinming Huang
^{1}

**2009**:251314

https://doi.org/10.1155/2009/251314

© W. Guo and X. Huang. 2009

**Received:**11 June 2009**Accepted:**8 September 2009**Published:**29 October 2009

## Abstract

Public safety organizations increasingly rely on wireless technology to provide effective communications during emergency and disaster response operations. This paper presents a comprehensive study on dynamic placement of relay nodes (RNs) in a disaster area wireless network. It is based on our prior work of mobility model that characterizes the spatial movement of the first responders as mobile nodes (MNs) during their operations. We first investigate the COverage-oriented Relay Placement (CORP) problem that is to maximize the total number of MNs connected with the relays. Considering the network throughput, we then study the CApacity-oriented Relay Placement (CARP) problem that is to maximize the aggregated data rate of all MNs. For both coverage and capacity studies, we provide each the optimal and the greedy algorithms with computational complexity analysis. Furthermore, simulation results are presented to compare the performance between the greedy and the optimal solutions for the CORP and CARP problems, respectively. It is shown that the greedy algorithms can achieve near optimal performance but at significantly lower computational complexity.

## Keywords

- Bipartite Graph
- Relay Node
- Mobility Model
- Disaster Area
- Binary Integer Programming

## 1. Introduction

In our prior work [1], we proposed a mobility model to describe the movement pattern of MNs within a large disaster area. Moreover, we studied the coverage problem with no capacity constraints on RNs. In this paper, we assume that each RN can support a limited number of users. Then the problem to be studied is formulated as finding the deployment of a given number of RNs such that: most MNs can be covered; the network throughput can be maximized.

We first study the COverage-oriented Relay Placement (CORP) problem of deploying a set of RNs to cover a maximum number of MNs. As an initial setup, we consider the subproblem of Relay Assignment for COverage-oriented Relay Placement (RA-CORP) which is, for any given RNs' placement, to obtain the optimal associations between RNs and MNs using maximal matching method. Secondly, the Greedy Incremental COverage (GICO) algorithm is proposed to iteratively find the optimal location for the RN, one at each time. Thirdly, we put forward the constrained exhaustive search (CES) method to produce the optimal solution to the CORP problem as a benchmark for the GICO algorithm.

We also investigate the CApacity-oriented Relay Placement (CARP) problem to maximize the total throughput of DAWN. In this case, the Relay Assignment for CApacity-oriented Relay Placement (RA-CARP) can be formulated as the assignment problem and solved by the Hungarian method [2]. Subsequently, we propose the Greedy Incremental Capacity (GICA) algorithm to find the RNs' positions one by one. In comparison, the optimal placement of all RNs can be obtained by solving a complicated binary integer programming problem but at very high computational complexity.

The rest of this paper is organized as follows. In Section 2, related work on disaster area networks, mobility model, and base station placement in wireless networks is summarized. In Section 3, we describe the mobility model of MNs within the disaster area; Section 4 presents the problem formulation of maximizing coverage for DAWN; Section 5 presents the technical approaches to solve the CORP problem. In Section 6, we present the problem formulation of maximizing aggregate throughput for DAWN. Subsequently, Section 7 presents the technical approaches to solve the CARP problem; simulation results are given in Section 8, followed by conclusions in Section 9.

## 2. Related Work

### 2.1. Disaster Area Wireless Network

Recently, many kinds of wireless networks have been proposed to be applied to disaster area relief operations. In [3], Hiroaki et al. propose and evaluate a mobile ad hoc network system to pursue the location and personal information of victims in occurrence of disaster. In [4], Kanchanasut et al. describe an emergency communication network platform designed for collaborative simultaneous emergency response operations using a combination of mobile ad hoc networking and a satellite IP network operating with conventional terrestrial internet. In [5], Malan et al. introduce a wireless infrastructure intended for emergency medical care, integrating low-power, wireless vital sign sensors, and PC-class systems. In addition, Zussman and Segall propose to construct an ad hoc network of wireless smart badges in order to acquire information from trapped survivors [6]. Besides, a novel ballooned wireless mesh network [7] has been proposed for emergency information system. All these works either assume the majority network nodes are static or mention little about their mobility model. Therefore, they fail in constructing the disaster area communication system to accommodate dynamic node configurations.

### 2.2. Macroscopic Mobility Model

In the recent years, several different macroscopic mobility models have been proposed and used for performance evaluation of networks. The fluid mobility model [8–10] conceptualizes traffic flow of users as the flow of a fluid, which models mobility in terms of the mean number of users crossing the boundary of a given area. Derived from transportation theory, these models give an aggregated description of the movement of several users, ranging from street scale and city scale [11–13] to national and international scales [12, 14]. Furthermore, two different event-driven role-based mobility models are designed for disaster area relief applications [15, 16]. However, these two models only apply in small area with specific disaster sites.

### 2.3. Base Station Placement

There have been extensive researches dedicated to base station placement problem in wireless sensor networks. In [17], a multiobjective metric is proposed for placing multiple base stations at the optimal positions in wireless sensor networks, including coverage, fault tolerance, energy consumption, and network delay. In [18], Shi and Hou propose a (1- ) optimal approximation algorithm to place base station so that the network lifetime could be maximized. In [19], a polynomial time heuristic is proposed for optimal base station selection within a wireless sensor network. In [20], Pan et al. study base station placement problem to maximize network lifetime. Most of existing base station placement schemes are designed for wireless networks with nodes at specific positions. Therefore, they are not suitable for the proposed mobile scenarios.

## 3. System Modeling and Applications

### 3.1. Mobility Model

### 3.2. Network Model

We consider a set of MNs moving within the disaster area following the mobility model described previously and assume that a fixed number of RNs are ready for deploying to connect all MNs to the backbone network. We assume that all MNs have small transmission range . The transmission region of an MN is defined as the area in which all points are within distance from the MN. The th MN can communicate bi-directionally with the th RN if the distance between them . In other words, is said to be covered by if is within the transmission region of . RNs are able to communicate with each other without distance constraints and they form the backbone network. We assume that the relay stations can be installed on vehicles and can quickly move to any locations in the disaster area. We assume each MN occupies one orthogonal channel associated with an RN for at least one time unit to communicate bi-directionally. Since the RN has a limited bandwidth, each RN can only support a certain number of MNs. As a note, no interference issues are considered in our network model due to abundant unoccupied spectrum in the disaster area.

## 4. Maximizing Node Coverage: Problem Formulation

We first provide the notation used in this section. Let denote the set of four vertices of square . denotes the disk centering at with radius as . A spot is said to be covered by if , denoted as . A polygon is said to be covered by if for any point within , , denoted as .

For the CORP problem, RNs should be placed at positions to connect the maximum number of MNs. As MNs follow a macroscopic mobility model, we choose to cover the active (busy) squares instead of tracking the individual MNs. In other words, if a busy square is covered, then all MNs within the square are connected. We now claim Theorem 1 to show that a square can be covered by a circle if and only if its four vertices are within that circle.

Theorem 1 (covering a polygon).

Assume a polygon , where and denote the set of vertices and edges, respectively. If for all vertex , , then .

Proof.

First of all, we need to acknowledge the fact that if , , then ( denotes the edge connecting and ), (it is obviously true since the edge is fully contained in one circle if the two terminals are within the same circle). For all , we have , where and . Since for all , , then and . As a result, , and .

*feasible circle*of each square. For every busy square in the DAWN, each has a corresponding feasible circle. These feasible circles may overlap each other and the intersected areas are referred as

*shared regions*. As shown in Figure 4, three feasible circles are intersected and an RN placed in the shared region (shaded area) can cover all 3 squares at the same time. In a DAWN, these shared regions form a candidate set for RN placement. In this paper, our analysis and simulations are derived directly based on the concept of shared regions, instead of using traditional Cartesian coordinates.

The CORP problem is defined as follows. Given a set of busy squares, in which each contains some MNs inside with transmission range , and RNs each with capability , find the optimal placement of the RNs, such that the maximum number of MNs is covered.

## 5. Maximizing Node Coverage: Technical Approaches

In this section, we first present a maximal matching method to solve the RA-CORP problem if the RNs' positions are known. Secondly, we propose the GICO and CES algorithms to tackle the CORP problem. In addition, we conduct complexity analysis for both algorithms.

### 5.1. Relay Assignment for Fixed RN Positions (RA-CORP)

The RA-CORP problem tries to find the optimal association between MNs and RNs. We use a bipartite graph to represent the RA-CORP problem, and then use a sparse matrix-based algorithm to find the maximum-sized matching between the MNs and RNs.

Definition 1.

Denote the feasible circle of the
th busy square
as
. Assume
,
have some area overlap, then the intersection area is defined as a *shared region* that only MNs within busy squares
(
) can access, denoted as
, or simply as
.

At any time, MNs are distributed within a set of busy squares. The feasible circles of these busy squares intersect and yield a set of shared regions where denotes the cardinality of the set . A fixed number of RNs are deployed at , . Each RN can support at most MNs within the squares covered by the RNs. Then the RA-CORP problem is formulated as

where denotes that is connected with and 0 otherwise. denotes the set of RNs that cover . denotes the set of MNs that are covered by . The second constraint demands that each MN can at most connect to one RN. The third constraint shows that at most MNs can connect to one RN.

In this paper, we use a sparse matrix-based approach [21] to find the maximal matching between MNs and channels of RNs for each RN. This approach yields the optimal solution. The complexity of finding the maximal matching within a bipartite graph is .

### 5.2. Relay Placement for Optimal Coverage (CORP)

We first claim Theorem 2 about complexity of the CORP problem.

Theorem 2.

The CORP problem is NP-complete.

Proof.

See Appendix .

Then we perform aggregation for all shared regions to reduce the solution space. Since the CORP problem is NP-complete, we introduce a heuristic approach GICO to solve the problem. To measure the performance of GICO, we also give the optimal solution by employing the CES algorithm.

#### 5.2.1. Aggregation

The aggregation procedure aims to reduce the cardinality of the set of shared regions, thus greatly reduces the solution space. Given a set of shared regions , the aggregation proceeds as in Algorithm 1. We say belongs to or contains if Let and denote the set of all shared regions, and reduced set of shared regions, respectively. denotes the cardinality of the set .

**Algorithm 1:** Aggregation procedure.

**Aggregation**(
)

- 2
**for**i=2 to - 3
sign=0;

- 4
**for**j=1 to - 5
**if**belongs to - 6
remove from ;

- 7
sign=1;

- 8
**end if**; - 9
**if**belongs to - 10
sign=2;

- 11
break;

- 12
**end if**; - 13
**end for**; - 14
**if**sign==0 or sign==1 - 15
add to ;

- 16
**end if**; - 17
**end for**; - 18
**return**;

The procedure reduces the cardinality of the set of shared regions by removing those that are contained by others. The procedure reduces the solution space without losing any coverage options. The reason is that RNs placed in that contains can cover busy squares . Therefore, the set can contain all shared regions but with the minimal cardinality.

#### 5.2.2. Greedy Incremental Coverage (GICO)

The GICO algorithm is based on the following idea. Although it is not computationally feasible to perform an exhaustive search for placing RNs simultaneously, it is possible to choose an optimal position to place one RN at a time. When the RN is placed at each shared region, the optimal relay assignment can be obtained by utilizing the maximal matching method. The best shared region for placing one node can be found by exhaustively searching all shared regions in . Once the location for this RN is fixed, the next RN can be placed following the same procedure. It should be noted that when placing next RN, those previously placed RNs should be jointly considered for relay assignment in order to compute the coverage values. In this approach, the RNs are placed one by one until all RNs are deployed in the set .

As listed in Algorithm 2, GICO works as follows. represents the shared regions that have been chosen to place RNs. denotes the cardinality of . It is possible that contains multiple same shared regions, which means that multiple RNs should be placed in that shared region. At line 4, each shared region in the set is added into the current set and the set is obtained. The procedure at line 5 calculates and stores the maximal matching based on RNs' deployment according to the current . denotes the calculated optimal maximal matching while denotes the set of shared regions that each contains one RN. After executing the procedure of lines 4 and 5 for times, the best next shared region to place one RN is found (line 7) and added to the set (line 8). Therefore, after greedily choosing RN placement one by one for times, we can finally obtain the solution.

**Algorithm 2:** GICO.

**GICO**( )

- 1
1 ;

- 2
**for**i=1 to - 3
**for**j=1 to - 4
- 5
**RA-CORP**( ) - 6
**end for**; - 7
[maximum, index]=

**Max**(value); - 8
;

9**end for**;

10 **return**
;

#### 5.2.3. Constrained Exhaustive Search (CES)

In order to obtain the optimal solution as a benchmark for our GICO algorithm, we need to search all possible combinations of the shared regions. However, even after employing the aggregation procedure to reduce the size of solution space, the complexity for searching the optimal solution could still be as high as Therefore, we resort to devising the CES algorithm to further reduce the solution space by adding one constraint to the combinations of shared regions. The constraint is that for each set of RN placements, each RN should cover at least one MN based on the RA-CORP results. In particular, the number of RNs placed in one shared region times RNs' capability should not exceed the total number of MNs in those busy squares that are covered by the RNs by more than , shown as

where denotes the number of RNs placed at and denotes the number of MNs in the busy square .

### 5.3. Complexity Analysis

We first discuss the complexity of the aggregation procedure. Based on Algorithm 1, the procedure between line 5 to line 12 is iterated times, and the procedure between line 14 to line 16 is iterated times. As neither nor is influenced by , the complexity for the aggregation procedure is .

Secondly, we analyze the complexity of GICO. Based on Algorithm 2, the complexity of the GICO algorithm is , because the procedure from line 4 to line 5 is iterated times, and the worst case complexity for the maximal matching method is .

For CES, the complexity analysis is more complicated. According to the constraint in (2), each shared region cannot host more than a limited number of RNs. Therefore, we assume on average each shared region can host ( ) RNs, as shown in

In other words, the list of shared regions can be extended to a longer one with length , on which each shared region can host at most one RN. As the worst case complexity for the maximal matching method is , the computation complexity for CES can be stated as . Now we claim that the complexity of the CES algorithm is higher than GICO, as shown in Theorem 3.

Theorem 3 (complexity of GICO and CES).

The complexity of the GICO algorithm is lower than the complexity of the CES algorithm.

Proof.

Obviously, the term . Therefore, the complexity for GICO is less than the complexity for CES.

## 6. Maximizing Aggregate Throughput: Problem Formulation

In Section 4, we formulate the CORP problem, which is aimed to maximize the number of MNs that can be connected to the backbone network. However, the objective of the CORP problem does not address the quality of service (QoS) requirements of individual links. In other words, the deployment of RNs has to consider not only the coverage but also the QoS performance with intelligent channel allocation. Therefore, we put forward the CARP problem in the interest of enhancing the QoS performance of DAWN.

where denotes the area of the polygon formed by connecting the intersection points in a counterclockwise order, which can be calculated using

Based on Shannon formula, the channel capacity of a link can be expressed as (8) using path loss model

where denotes the capacity of the channel, denotes the bandwidth of the channel, denotes the distance between the transmitter and receiver of the link, denotes the path loss coefficient, denotes the transmit power, denotes the noise power, and are the transmitter and receiver antenna gains, respectively, denotes the wavelength of the transmitted signal, denotes the frequency of the transmitted signal, is the velocity of radio-wave propagation in free space, which is equal to the speed of light.

Since MNs follow the macroscopic mobility model, we resort to developing the two-dimensional integral (9) to compute the throughput of the link between an MN and its assigned RN. In (9), denotes the coordinates of the left downward vertex of the th busy square . denotes the throughput of the link between one MN in and one RN placed at . denotes the side length of each square

The CARP problem is formulated as follows. Given MNs each with transmission range that are distributed within a set of busy squares and RNs each with capability , find the optimal positions for the RNs such that the aggregated throughput of all established links between MNs and RNs are maximized.

## 7. Maximizing Aggregate Throughput: Technical Approaches

In this section, we investigate the CARP problem of deploying a set of RNs to maximize the total throughput of DAWN. We first consider the optimal relay assignment for fixed RN positions, which can be solved using the Hungarian method. On this basis, we propose the GICA approach to tackle the CARP problem. In addition, since the CARP problem falls into a binary integer programming formulation, the branch and bound algorithm [22] is adopted to produce the optimal solution as the benchmark for the GICA approach.

### 7.1. Relay Assignment for Fixed RN Positions (RA-CARP)

At any time, MNs are distributed within a set of busy squares. The feasible circles of these busy squares intersect and yield a set of shared regions A fixed number of RNs are deployed at the set of centroid points . Each RN can support at most MNs to access the network in the squares that it covers. Now the RA-CARP problem is formulated as

where denotes that is connected with and 0 otherwise; denotes the index of the busy square where is. denotes the index of the shared region where is placed. The second constraint denotes that each MN can at most connect to one RN. The third constraint shows that at most MNs can connect to one RN.

Given MNs placed within busy squares and RNs deployed in some shared regions, the bipartite graph can be generated as in Figure 5. It can be seen that for each MN-RN pair, there are edges each with a weight equal to the capacity value of the corresponding link. Note that for those pairs that the RN does not cover the MN, the edges are assigned weights equal to 0. We now can generate a gain matrix shown as

where and . We now present the Hungarian method [2] as follows.

Step 1.

If is not a square matrix, we have to augment into a square matrix by padding rows or columns with all zeros.

Step 2.

Multiply the matrix by .

Step 3.

Subtract the minimum value of each row from row entries.

Step 4.

Subtract the minimum value of each column from column entries.

Step 5.

Select rows and columns across which you draw lines, such that all zeros are covered and that no more lines have been drawn than necessary.

Step 6.

If the number of lines equals the number of rows, choose a combination of zero elements from the modified gain matrix such that the position of each chosen element is incident on a unique row and column. Then the optimal assignment result consists of the RN-MN pairs as represented by the chosen elements in the modified gain matrix. If the number of lines is less than the number of rows, go to Step 7.

Step 7.

Find the smallest element which is not covered by any of the lines. Then subtract it from each entry which is not covered by the lines and add it to each entry which is at the intersection of a vertical and horizontal line. Go back to Step 5.

### 7.2. Relay Placement for Maximal Aggregate Throughput (CARP)

We claim Theorem 4 about complexity of the CARP problem.

Theorem 4.

The CARP problem is NP-complete.

Proof.

See Appendix .

Since the CARP problem is NP-complete, we introduce a heuristic approach GICA to solve the problem. To measure the performance of GICA, we also present the optimal method for the CARP problem.

#### 7.2.1. Greedy Incremental Capacity (GICA)

According to Algorithm 3, the algorithm works as follows. denotes the set of centroid points that have been chosen to place RNs. denotes the set of centroid points of all shared regions. denotes the set of centroid points that have been chosen. denotes a current set of centroid points with each hosting one RN. At line 4, each centroid point in is added to and the set is obtained. The procedure at line 5 calculates and stores the total throughput yielded by the Hungarian method based on . denotes the calculated optimal association between MNs and RNs. After executing the procedure of lines 4 and 5 for times, the best next centroid point to place one RN is found (line 7) and added to (line 8). Therefore, after greedily choosing centroid points one by one for times, we can finally obtain the set of centroid points to place RNs.

**Algorithm 3:** GICA.

**GICA**(
)

- 1
;

- 2
**for**i=1 to - 3
**for**j=1 to - 4
- 5
**RA-CARP**( ) - 6
**end for**; - 7
[maximum, index]=

**Max**(value); - 8
;

- 9
**end for**; - 10
**return**;

#### 7.2.2. Optimal Solution to CARP Problem

We show that the RA-CARP can be formulated as a binary integer programming problem when RNs are placed at fixed positions. Subsequently, the GICA method utilizes the Hungarian method to greedily place one RN at an iteration. Therefore, the solutions yielded by GICA cannot be guaranteed optimal because the RN assignment and placement are considered separately. It would be natural to believe that only when we search all solution space can the optimal solution be produced.

Hereby we introduce two binary variables and denotes that is connected with and 0 otherwise; denotes that is placed at the centroid point of the th shared region and 0 otherwise. Then we can jointly formulate the CARP problem as

Since the objective term contains the product of two variables and , it is difficult to solve. According to [23], the product of multiple binary variables can be substituted by a new variable with two constraints that ensure that if there exists , and if for all . Therefore, we transform (12) into a binary integer programming problem shown as

Now the CARP problem is formulated as a binary integer programming problem. We then utilize the branch and bound algorithm to solve it. The algorithm searches for an optimal solution by solving a series of LP-relaxation problems, in which the binary integer requirement on the variables is replaced by the weaker constraint . More details can be referred to [22].

### 7.3. Complexity Analysis

We first discuss the computation complexity of the Hungarian method to assign MNs when RNs are placed at fixed positions. According to [2], the complexity is .

Then we analyze the complexity of the GICA algorithm. Let denote the number of shared regions. Based on Algorithm 3, as the procedure on lines 4 to 5 is iterated times, the computation complexity of the GICA algorithm is .

The optimal method to the CARP problem uses the branch and bound algorithm to solve a binary integer programming problem. As the number of binary variables is , the worst case complexity for the optimal method is . It is apparent that the complexity of the GICA algorithm is much lower than the optimal algorithm.

## 8. Simulation Results

In this section, we present the numerical results obtained from the simulation using high level programming language. For the CORP problem, we compare the performance of the GICO algorithm and the CES algorithm. For the CARP problem, we compare the performance of the GICA algorithm and the optimal algorithm. It is illustrated that the two greedy algorithms both merit close-to-optimal performance and low complexity.

### 8.1. Simulation Setup

The initial configuration of CI values of squares in disaster area. The CI values are randomly chosen between the interval .

Square index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 10 | 10 | 5 | 9 | 2 | 5 | 10 | 8 | 10 |

2 | 7 | 1 | 9 | 10 | 7 | 8 | 8 | 4 | 7 | 2 |

3 | 8 | 1 | 3 | 1 | 1 | 9 | 7 | 4 | 10 | 1 |

4 | 5 | 4 | 8 | 8 | 2 | 5 | 5 | 7 | 8 | 8 |

5 | 3 | 7 | 7 | 2 | 2 | 5 | 10 | 4 | 6 | 3 |

6 | 8 | 3 | 6 | 7 | 9 | 10 | 6 | 2 | 2 | 3 |

7 | 9 | 3 | 9 | 3 | 10 | 4 | 2 | 3 | 7 | 5 |

8 | 4 | 9 | 6 | 6 | 10 | 3 | 8 | 8 | 4 | 6 |

9 | 1 | 1 | 6 | 8 | 10 | 2 | 6 | 5 | 1 | 4 |

10 | 2 | 8 | 4 | 6 | 2 | 7 | 3 | 7 | 7 | 8 |

### 8.2. Simulation Results for Maximal Coverage

We introduce *coverage percentage* as the measurement of coverage performance, which is defined as the ratio of the number of covered MNs to the total number of MNs working in the disaster area.

### 8.3. Simulation Results for Maximal Throughput

## 9. Conclusion

In this paper, we study the dynamic deployment of mobile relays in DAWN to enable and improve the communications for the first responders during their operations. A mobility model is used to capture the movement pattern of the MNs and their communications to the RNs. Given a fixed number of relay nodes, the optimization problem is to determine the locations of the RNs as the MNs move in the disaster area nomadically. Two performance objectives, including maximal node coverage and maximal network capacity, are considered, respectively, in this study.

In the coverage problem (CORP), the performance objective is to place the RNs that can connect the maximum number of MNs in the network. As a preliminary step, we employ a maximal matching method to find the optimal relay node assignment for the static network scenario, that is, all RNs are fixed. Subsequently, we present the greedy incremental coverage algorithm (GICO) and the optimal constrained exhaustive search (CES) algorithms. The GICO algorithm is suboptimal but with significantly less computational complexity than the CES algorithm. The simulation results show that GICO algorithm can achieve close to optimal performance at different network setup and configurations.

In the capacity problem (CARP), the performance objective is to maximize the aggregated network throughput for all MNs in the DAWN. As an initial step, we first consider the relay node assignment for the static case that can be solved using the Hungarian method. Similarly, we also present both the greedy incremental capacity algorithm (GICA) and the optimal algorithm. The optimal solution for CARP can be obtained through the binary integer programming approach but at much higher computational complexity. The simulation results show that the GICA algorithm can produce near optimal results.

In addition, it is observed that network generally yields better coverage and throughput performance for scenarios in which all MNs start from 1 corner than from 4 corners of a disaster area. The tradeoff is that it would require longer time to clear the entire disaster area. As a conclusion, we advocate using the greedy algorithms to determine the dynamic relay placement in the deployment of disaster area wireless networks, in which real-time computation is practically more important.

## Appendices

### A. Proof of Theorem 2

NP-completeness applies directly not to optimization problems, however, but to decision problems, in which the answer is simply "yes" or "no" [24]. We first present the decision problem CORP-D associated with the CORP problem as follows. Given a set of busy squares, the number of MNs in each square, the transmission range and RNs each with capability , is it possible to cover all MNs using exactly RNs? To prove the NP-completeness of the CORP problem, it suffices to prove that the decision problem CORP-D is NP-complete.

We start by arguing that CORP-D NP. Then we prove that the CORP-D problem is NP-hard by showing that MSC CORP-D, ( denotes a transformation of polynomial time. MSC denotes the NP-complete minimum set cover problem). Because the CORP-D problem is both NP and NP-hard, it is NP-complete.

To show CORP-D NP, we deploy RNs in the shared regions. Then to find if such deployment of RNs can cover all MNs is tantamount to solving the RA-CORP problem. As the RA-CORP problem has been proved to be solved in polynomial time, CORP-D NP.

We next prove that MSC CORP-D, which shows that CORP-D is NP-hard. Let be an instance of the problem, where denotes the collection of subsets of a set , denotes the minimum cardinality of the set such that and . To obtain an instance of the CORP-D problem we only need to define the capacity bound for each RN. Let be the number of all MNs. Then we build the relationship between instances of the CORP-D problem and the MSC problem as follows. Each element in the set corresponds to one MN; each shared region corresponds with one subset . is covered by the th shared region if . Then we must prove that RNs can cover all MNs if, and only if, there exists , such that and .

First, suppose that MNs in a set of busy squares can be covered by RNs, each with a capacity of . Then for deployed at one shared region, the corresponding subset is chosen as one element in . Since RNs can cover all MNs, it is easy to see that and .

Now assume that there exists such that and . For we can place one RN at the th shared region. In the meantime, covers if . Since all MNs are covered by the RNs. Since , the number of RNs placed is no larger than .

Thus we have shown that the CORP-D problem is NP-complete, which completes the proof.

### B. Proof of Theorem 4

We start by arguing that CARP-D NP (CARP-D is the decision problem associated with the CARP problem). Then we prove that CARP-D is NP-hard by showing that CORP-D CARP-D. Because the problem CARP-D is both NP and NP-hard, the problem CARP is NP-complete.

To show CARP-D NP, we deploy RNs in the shared regions. Then to find if such deployment of RNs can produce the objective amount of throughput is tantamount to solving the RA-CARP problem. As the RA-CARP problem has been proved to be solved in polynomial time, the problem CARP-D NP.

We next prove that CORP-D CARP-D, which shows that the problem CARP-D is NP-hard. Let and a positive integer be an instance of the CORP-D problem. To obtain an instance of the CARP-D problem we only need define the capacity of each link between one MN and RN. Let the capacity for every link between each RN and MN be 1 unit. Then we recognize the instance of the CARP-D problem the same as the instance of the CORP-D problem. This transformation surely consumes polynomial time. Subsequently, we must prove that units of throughput can be produced if and only if MNs can be covered by the same set of RNs.

First, suppose that MNs in a set of busy squares can be covered by RNs. Then for covered by , one link is established between and to produce one unit of capacity. Since MNs are covered, it is easy to see that units of throughput can be produced.

Now we assume that the overall network capacity is units. Then for each link between and , we render cover . As there are links available, each associated with one MN, MNs can be covered by the same set of RNs.

Thus we have shown that the problem CARP-D is NP-complete, which completes the proof.

## Declarations

### Acknowledgments

This work has been supported in part by the National Science Foundation (NSF) through Award ECS-0725522 and by the Faculty Advancement in Research Awards from WPI. This work was presented in part at 3rd Intl. Conf. Wireless Algorithms, Systems and Applications, 2008.

## Authors’ Affiliations

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