On the number of channels required for interferencefree wireless mesh networks
 Aizaz U Chaudhry^{1}Email author,
 John W Chinneck^{1} and
 Roshdy HM Hafez^{1}
https://doi.org/10.1186/168714992013229
© Chaudhry et al.; licensee Springer. 2013
Received: 12 December 2012
Accepted: 2 August 2013
Published: 14 September 2013
Abstract
We study the problem of achieving maximum network throughput with fairness among the flows at the nodes in a wireless mesh network, given their location and the number of their halfduplex radio interfaces. Our goal is to find the minimum number of nonoverlapping frequency channels required to achieve interferencefree communication. We use our existing Select x for less than x topology control algorithm (TCA) to build the connectivity graph (CG), which enhances spatial channel reuse to help minimize the number of channels required. We show that the TCAbased CG approach requires fewer channels than the classical approach of building the CG based on the maximum power. We use multipath routing to achieve the maximum network throughput and show that it provides better network throughput than the classical minimum powerbased shortest path routing. We also develop an effective heuristic method to determine the minimum number of channels required for interferencefree channel assignment.
Keywords
1. Introduction
In the classical approach to channel assignment in multiradio multichannel (MRMC) wireless mesh networks (WMNs) [1–7], the number of available nonoverlapping frequency channels is assumed to be fixed. In such schemes, two links that are within the interference range of each other could be assigned the same frequency, causing cochannel interference which degrades the network throughput.
We study the channel assignment problem in MRMC WMNs from a different perspective. First, we ensure interferencefree communication among the mesh nodes by ensuring that links within the interference range of each other are assigned different channels. Second, instead of working with a fixed number of orthogonal frequency channels, we search for the minimum number of channels that provides an interferencefree channel assignment which allows the mesh network to achieve the maximum throughput.
We assume a single mesh gateway (GW). All mesh nodes, except the GW, are sources of flow. The GW is the sink for all flows. We formulate the MRMC WMN routing problem as a mixed integer linear program (MILP) whose objective is to maximize the network throughput while maintaining fairness among the multiple flows subject to flow conservation, halfduplex, and nodedegree constraints. The problem of assigning channels to the links involved in routing so that communication among mesh nodes is interference free is similar to the minimum coloring problem for the conflict graph, which is known to be NPhard for general graphs [8]. This amounts to finding the smallest number of maximal independent sets (MaISs), where the number of channels required is equal to the number of MaISs. To minimize the number of channels required, we use our existing Select x for less than x topology control algorithm (TCA) [2] to build the connectivity graph (CG). By controlling network connectivity, it lowers the transmitted power as much as possible without sacrificing network throughput and so tends to reduce transmitted power throughout the network, which supports green networking for WMNs.
Specifically, the contributions of our work are as follows.

We develop a method to determine the minimum number of nonoverlapping frequency channels required for interferencefree channel assignment given the locations of the mesh nodes and the number of their halfduplex radio interfaces. This is also the number of channels required to achieve the maximum network throughput^{a}.

We show that our TCAbased approach for building the CG outperforms the classical maximum powerbased CG approach for all nodedegree constraints in terms of the number of channels required as well as the linkstochannels ratio.

We show that the multipath routing approach significantly outperforms the minimum powerbased shortest path (MPSP) routing approach in terms of network throughput at higher nodedegree constraints.

We develop and compare two effective new heuristics for interferencefree channel assignment (CA).
The rest of the paper is organized as follows. Section 2 presents related work. Our model for the network architecture is given in Section 3. Section 4 explains the creation of the connectivity graph using the two different approaches. The problem formulation for multipath routing and MPSP routing are also presented. The creation of the conflict graph using the protocolbased interference model is given in Section 5, which also presents two heuristic approaches for the solution of the channel assignment problem. Performance evaluation with results is given in Section 6. Conclusions and some directions for future work are given in Section 7.
2. Related work
There is a great deal of literature on improving the performance of MRMC WMNs by maximizing the network throughput [1–7]. In all of these studies, the number of available nonoverlapping frequency channels is assumed to be fixed, so links within the interference range of each other could be assigned the same channel, causing cochannel interference that degrades network throughput. We provide interferencefree communication among the mesh nodes by ensuring that links that fall within the interference range of each other are assigned different nonoverlapping frequency channels. We then determine the minimum number of channels required to realize such interferencefree channel assignment in order to achieve the maximum network throughput.
In addition to channel assignment, the schemes in [1–7] also deal with routing in MRMC WMNs. In the previous work [1, 2], we employed minimum powerbased minimum spanning trees and minimum powerbased shortest path trees for degree constrained routing. In [3], traffic is routed using either minimumhop path routing or randomized multipath routing. In [4], routing heuristics incorporate the impact of interface switching cost and a possible implementation using the dynamic source routing protocol is discussed. The optimized link state routing protocol is used in [5] for route selection. The scheme in [6] uses a flow rate computation method for routing, which aims to maximize the network throughput. For the performance evaluation of the channel assignment scheme in [7], the routes are computed statically using the smallest number of hops. In this paper, we formulate the routing problem as a MILP with the objective of maximizing the network throughput under fairness, flow conservation, halfduplex, and nodedegree constraints.
The schemes in [2–7] focus on improving the throughput in MRMC WMNs without considering fairness. In [1], we proposed channel assignment algorithms to improve throughput as well as fairness. The scheme in [9] achieves a good tradeoff between throughput and fairness even though it does not find an absolutely even distribution. The authors have shown in [10] that the network throughput as well as fairness increases as the number of available radio interfaces per router or the number of available orthogonal frequency channels increases. The approach in [11] deals with congestion control and channel assignment and achieves significant gains in terms of network utilization and establishing fairness. In [12], the authors deal with the problem of joint channel assignment, link scheduling, and routing for throughput optimization, and show that the fairness and throughput achieved by their method is within a constant factor of the optimum value.
An algorithm is proposed in [13] for joint channel, capacity, and flow assignment in MRMC WMNs. It first tries to maximize the fairness and then uses the remaining unused network resources to maximize the overall network throughput. The divideandconquer approach in [14] splits the joint routing and channel assignment problem into separate subproblems. This significantly improves fairness among the traffic flows. Throughput and fairness do not normally go hand in hand, and increasing one generally decreases the other. Our proposed approach, however, achieves maximum network throughput for MRMC WMNs and at the same time ensures fairness among the network flows.
Algorithms for solving the maximum independent set (MIS) problem have been widely used for resource allocation in multihop wireless networks. The scheduling scheme in [15] uses independent sets for feasible link scheduling in TDMAbased WMNs. The method in [9] uses maximal independent sets for link scheduling in multichannel WMNs. In [16], the authors propose a polynomial time approximation scheme for computing an independent set from the link interference graph, as large as (1  ϵ) times the cardinality of the MIS. Given the number of channels among other inputs, the method is then used to develop a channel assignment for MRMC WMNs such that the number of links in the communications graph that can be active simultaneously is maximized. In [17], maximal weighted independent set solutions are used to develop an algorithm for link scheduling in multiradio multichannel multihop wireless networks. A polynomial computing method in [18] searches for the critical maximal independent set that needs to be scheduled for optimal resource allocation. In [19], the authors use solutions for the maximum weighted independent set problem to develop approximation algorithms for link scheduling, and compute a maximum (concurrent) multiflow in multiradio multichannel multihop wireless networks. We solve the minimum coloring problem heuristically by repeatedly solving the MaIS subproblem to determine the minimum number of channels required for interferencefree CA in MRMC WMNs.
3. Network architecture
We assume that each mesh node is equipped with multiple radio interfaces. One of these radios is used for control traffic, while the others are used for data traffic. We define the nodedegree of a mesh node as the number of neighbors with which it can communicate data traffic simultaneously. For example, a nodedegree of 2 means that each mesh node is equipped with two radio interfaces for data traffic and can communicate with at most two of its neighbors simultaneously.
The radio interfaces are assumed to be halfduplex; hence, a mesh node cannot send and receive at the same time using the same radio interface. It is assumed that each radio interface of a multiradio mesh node is equipped with an omnidirectional antenna and that the radio interfaces of nodes can be tuned to different nonoverlapping frequency channels. The control radios of all nodes are tuned to a common frequency channel for communication of the control traffic.
4. Routing problem
4.1. Connectivity graph
We compare two different approaches for building the CG C(V,E), where vertices V c orrespond to the wireless nodes and the edges E correspond to the wireless links between the nodes. The first approach is the classical way based on maximum transmission power, while the second approach uses our Select x for less than x TCA.
The Select x for less than x TCA builds a CG using topology control to mitigate the cochannel interference and enhance spatial channel reuse while preserving network connectivity. Each mesh router (MR) broadcasts a Hello message containing its node ID and position over the control channel using the control radio at maximum power. From the information in the received Hello messages, each MR arranges its neighboring nodes in ascending order of their distance. The result is the maximum power neighbor table (MPNT). Then, each MR sends its MPNT along with its position and node ID to the GW over the control channel. For each MR in the network, the GW builds a direct neighbor table (DNT) by selecting at least x nearest nodes for that MR. If required, the GW then converts some unidirectional links in the DNT of a mesh node into bidirectional links, which results in the final neighbor table (FNT) [2]. Bidirectional links are required for linklevel acknowledgments and to ensure the existence of reverse paths.
We build the CG for the maximum power (MP) based approach using the MPNT and the CG for the TCAbased approach using the FNT. Node locations are assumed to be known. In order to achieve a strongly connected topology, we assume a maximum transmission range of 164 m for all mesh nodes.
4.2. Problem formulation for multipath routing
We formulate the multipath routing problem in multiradio WMNs as a MILP. We call the index p ∈ P a commodity. Let

P be the commodities, i.e., sourcedestination pairs (s_{ 1 }, t_{ 1 }),.…., (s_{ P }, t_{ P });

f_{ ij }^{ p } be a variable denoting the amount of flow of commodity p on link l_{ ij };

f_{ s }^{ p } be a variable denoting the amount of inflow of commodity p from the source of p;

f_{ d }^{ p } be a variable denoting the amount of outflow of commodity p from the sink of p;

c_{ ij } be an input parameter denoting the capacity of link l_{ ij } where l_{ ij } ∈ E;

z_{ ij } be a binary variable such that z_{ ij } ∈ {0, 1} is 1 when the link l_{ ij } is used for routing and 0 otherwise;

dc be an input parameter denoting the constraint on the nodedegree of the mesh routers, such that dc ∈ {2,3,4,5,6};

cost1_{ ij } be an input parameter containing a cost of 0.0001 for each link l_{ ij };

demand_{ sd } be an input parameter representing flow demands between the sourcesink pairs and is equal to 1 for all commodities; and

y be a variable denoting the multiplier on the unit flow demand of the commodities.
4.2.1. Objective
The small value of cost1_{ ij } in (1) prevents redundant flowloops and does not affect the result.
4.2.2. Constraints
where i ∈ V \ {s_{ p }, d_{ p }} and p ∈ P.
where i ∈ V.
Varying x in the Select x for less than x TCA for different nodedegree constraints ensures that the total amount of flow in the network increases equally for the TCAbased and MPbased CG approaches with an increase in the nodedegree constraint. It creates a certain amount of connectivity in the CG for a certain nodedegree constraint and hence a certain number of links for the GW. For example, for nodedegree constraints of 2 and 3, we use the Select 3 for less than 3 TCA to ensure at least three links for the GW in the CG; for the nodedegree constraint of 4, we use the Select 4 for less than 4 TCA to ensure at least four links for the GW in the CG, and so on. Since the Select 2 for less than 2 TCA mostly leads to a disconnected network in the case of random and controlled random topologies, we use the Select 3 for less than 3 TCA for the nodedegree constraint of 2. The amount of total flow in the network depends on the number of links for the GW (sink) node for a given nodedegree constraint. For example, if the capacity of each link is 24, then the maximum possible total network flow, i.e., the maximum network throughput, is 48 for a nodedegree constraint of 2.
where s ∈ {sources}, d ∈ {sinks}, and p ∈ P.
We use the AMPL language [20] to model the multipath routing problem and IBM CPLEX 12.2 [21] to solve the resulting problem.
4.3. Problem formulation for MPSP routing
We formulate the MPSP routing problem in two stages. In the first stage, shortest paths are determined between the sourcesink pairs. The metric for path selection is minimum power. For the two CG approaches, the gateway calculates the minimum power required to reach each of the nodes in the FNT or the MPNT of a node by using the appropriate propagation model. The free space model is used for short distances, and the tworay ground reflection model is used for longer distances, depending on the value of the Euclidean distance in relation to the crossover distance [22]. If the distance between two nodes u and v is less than or equal to the crossover distance, i.e., d(u,v) ≤ cross_over_dist, the free space model is used, whereas if d(u,v) > cross_over_dist, the tworay model is used. In the second stage, the flows of individual commodities are maximized equally so as to achieve the maximum total flow in the network, using the shortest paths determined in the first stage.
4.3.1. First stage
We formulate this stage as the following MILP. Let

cost2_{ ij } be an input parameter containing the cost of each link l_{ ij } which is minimum power;

supply_{ i } be an input parameter which is 1 for sources; and

demand_{ i } be an input parameter which is −1 for the sink (gateway).
4.3.1.1. Objective
4.3.1.2 Constraints
where i ∈ V \ {s_{ p }, d_{ p }}.
The nodedegree, halfduplex, and capacity constraints are exactly the same as in (5), (7), and (8), respectively. Note that because of the integer demand and supply flows and the unimodularity property of the network matrix, the continuous flows f_{ ij }^{ p } will have binary values in the solution.
4.3.2. Second stage
We formulate this stage as the following linear program:
4.3.2.1 Objective
The additional term in (1) is not required in (15) since there is no possibility of flowloops due to preestablished shortest paths.
4.3.2.2. Constraints
As in the case of multipath routing, we use the AMPL language to model the MPSP routing problem and IBM CPLEX 12.2 to solve the resulting problem.
5. Channel assignment problem
5.1. Conflict graph
We use the protocolbased interference model [23] to build the conflict graph, which is widely used for modeling interference in wireless networks [11, 12, 18, 19]. The input to the conflict graph consists of the links involved in routing, i.e., the output of the routing problem, and the node locations.
Let d_{ ij } denote the distance between nodes n_{ i } and n_{ j }, R_{ i } be the transmission range of node n_{ i }, and R_{ i }' be the interference range of node n_{ i }. In the conflict graph F, the vertices correspond to the links in the connectivity graph C. An edge between the vertices l_{ ij } and l_{ pq } in F indicates that the links l_{ ij } and l_{ pq } in C cannot be active simultaneously. Note that links l_{ ij } and l_{ pq }, which are involved in routing, are bidirectional links at the MAC and physical layer levels. So, a link l_{ ij } which is involved in routing is checked eight times with every other link that is involved in routing while building the conflict graph. An edge is drawn between the vertices l_{ ij } and l_{ pq } if any of the following is true:

d_{ ip } ≤ R_{ i }' or d_{ iq } ≤ R_{ i }' or d_{ jp } ≤ R_{ j }' or d_{ jq } ≤ R_{ j }' or

d_{ pi } ≤ R_{ p }' or d_{ pj } ≤ R_{ p }' or d_{ qi } ≤ R_{ q }' or d_{ qj } ≤ R_{ q }'.
While building the conflict graph, we assume that the interference range is twice the transmission range.
5.2. Maximum independent set problem and the minimum coloring problem
An independent set of a graph G is a subset of vertices of G such that none of the vertices in the subset share an edge. In the conflict graph F, an independent set of vertices indicates a set of links in C which can be active simultaneously. An independent set is said to be maximal if it is not a subset of any larger independent set or maximum if there is no larger independent set in the graph (i.e., it is an independent set with maximum cardinality). The MIS problem consists of finding the largest subset of vertices of a graph such that none of these vertices are connected by an edge.
A coloring of the conflict graph F is an assignment of colors (channels) to vertices such that adjacent vertices receive different colors. The minimum coloring problem is the problem of computing a coloring of the vertices in the conflict graph F using as few distinct colors as possible; this is the same as the problem of finding the minimum number of channels to use such that there is no interference. The minimum coloring problem is well known to be NPhard for general graphs [8]. A greedy heuristic for its solution consists of these steps: (1) find a maximal independent set of vertices and assign the members of this set to the same channel, (2) remove these vertices from the conflict graph, and (3) repeat until all vertices are colored (assigned a channel). The number of channels required to achieve interferencefree communication among the mesh nodes is equal to the number of MaISs.
Since finding a maximum independent set is itself NPhard [24], we also use heuristic algorithms for this step of the solution to the interferencefree channel assignment problem.
5.3. MaISbased heuristics for CA
We use three greedy heuristic algorithms to find MaISs. Algorithm 1 Maximum nodedegree start selects a vertex from the conflict graph with the maximum nodedegree and introduces that vertex into the maximal independent set under construction. The algorithm then checks the other vertices of the conflict graph and puts them in the set if they do not have an edge with the vertices already in the set. The worstcase computational complexity of Algorithm 1 to find a MaIS is O(m^{ 2 }) where m is the number of nodes in the conflict graph.
Algorithm 1 Minimum nodedegree start starts by selecting a vertex from the conflict graph with the minimum nodedegree. Note that if the conflict graph has multiple vertices with the maximum nodedegree (Algorithm 1 Maximum nodedegree start) or with the minimum nodedegree (Algorithm 1 Minimum nodedegree start), then the starting vertex is selected randomly from among them. After removing a MaIS from the conflict graph, ties for the starting vertex in the revised conflict graph are again broken randomly if required. Algorithm 1 Random start starts by selecting a vertex at random from the conflict graph.
These randomized heuristics find MaISs very quickly but may return different results each time, so we run them multiple times. For a given topology of MRs, we run each of the three algorithm variants 25 times on the conflict graph and take the best solution over all 75 runs.
5.4. MISbased heuristics for CA
The greedy minimum coloring heuristic presumably works best if the maximum independent set can be found at each step instead of just a maximal independent set. We tested Wilf's algorithm [25] for finding the maximum independent set of a given graph G. It starts by choosing a vertex v* from the graph which has the highest nodedegree. If v* has at least two neighbors, the computational complexity of Wilf's algorithm to find a MIS in the given graph is O(1.47^{ m }), which reduces to O(1.39^{ m }) if v* has at least three neighbors, where m is the number of nodes in the graph. After selecting v*, the algorithm builds two sets. The set n1 contains all the nodes of the graph except v* and the set n2 excludes both v* and N(v*), the neighborhood of v*, i.e., the nodes that are connected to v* by an edge. maxset is the set which has the higher cardinality of the two sets. The vertices in maxset (and their incident edges) are deleted from G, and the procedure is repeated until G has no edges. The MIS consists of the vertices in the disconnected G. The pseudocode of our procedure using Wilf's algorithm is shown in Algorithm 2.
Note that the procedures related to routing and channel assignment are carried out at the GW, which then sends each MR its channel assignment and routing information over the control channel using the control radio. Based on the channel assigned to an MR to communicate with a neighbor and its distance to that neighbor, each MR applies power control and adjusts its transmission power accordingly.
6. Performance evaluation
We evaluate the performance of the TCAbased vs. MPbased CG approach, multipath vs. MPSP routing, and the MaISbased vs. MISbased heuristics for CA, based on network throughput, fairness, solution time of the routing stage in clock seconds, number of channels required (NCR), and linkstochannels ratio (LCR), i.e., the ratio of the number of links involved in routing to the number of channels required. Note that an average LCR of 2 for a topology means that on the average, two links are assigned the same channel for that topology, so a higher LCR indicates better performance.
6.1. Network topology
A controlled random topology (CRT) is used for the evaluation. A 500 m × 500 m physical terrain is divided into cells, and an MR is placed randomly within each cell using a uniform random distribution. Twentyfive different CRTs consisting of 36node networks are considered. Irrespective of its location, node 15 is the GW for all CRTs. All mesh nodes, except the GW, are sources of flow. As stated earlier, the GW is the sink for all flows.
6.2. Experimental results
6.2.1. TCAbased vs. MPbased CG approach
We compare the performances of the TCAbased vs. MPbased CG approaches for different nodedegree constraints. For both CG approaches, we use multipath routing in combination with the MaISbased heuristic for channel assignment. All CPLEX solver parameters are at their default settings, except mipgap = 0.01. This speeds the MILP solution at the possible expense of a small degradation in the objective value, though we found the optimum in all cases. For the TCAbased CG approach, if the solver finds a suboptimum solution for the multipath routing problem for nodedegree constraints of 2 or 3, we move to a higher TCA to build the CG, i.e., Select 4 for less than 4 TCA or Select 5 for less than 5 TCA, until an optimum solution is found. Note that for a nodedegree constraint of 2, the maximum value of total network flow (maximum network throughput) is 48; for a nodedegree constraint of 3, it is 72; and so on.
We collected mean values and statistics on the 95% confidence intervals for all measures. If the 95% confidence interval is tightly grouped around the mean, then it is not reported in a table. Most measures are graphed so that trends are immediately apparent.
6.2.1.1. Fairness
Total flow (TCAbased CG vs. MPbased CG approach)
Nodedegree constraint  CG approach  y  Total flow 

2  MPbased  1.3714  48 
TCAbased  
3  MPbased  2.0571  72 
TCAbased  
4  MPbased  2.7428  96 
TCAbased  
5  MPbased  3.4285  120 
TCAbased  
6  MPbased  4.1142  144 
TCAbased 
6.2.1.2. Network throughput
The results in Table 1 indicate that the total flow in the network (the flow reaching the gateway (sink)) increases equally for the MPbased and TCAbased CG approaches with an increase in the nodedegree constraint. For a nodedegree constraint of 2 in Table 1, y is 1.3714. The total flow in the network, i.e., the network throughput, is equal to the number of sources times y, i.e., 35 × 1.3714 = 48, for both CG approaches, which is equal to the maximum total network flow (maximum network throughput) for a nodedegree constraint of 2. In fact, the network throughput achieved is maximum for all nodedegree constraints for both CG approaches, as shown in Table 1.
Our centralized approach leads to an optimum solution in terms of network throughput while also ensuring fairness among the network flows since the required global network information is available at the GW.
6.2.1.3. Solution time of routing stage
Solution time (TCAbased CG vs. MPbased CG approach)
Nodedegree constraint  CG approach  Solution time (s)  95% CI for solution time 

2  MPbased  656.33  296.431,016.22 
TCAbased  138.24  102.00174.49  
3  MPbased  11.31  10.3212.30 
TCAbased  3.04  2.623.45  
4  MPbased  10.89  9.8311.96 
TCAbased  3.70  3.134.28  
5  MPbased  9.74  8.3211.17 
TCAbased  3.61  2.804.43  
6  MPbased  8.82  7.2610.38 
TCAbased  3.99  3.104.89 
6.2.1.4. Number of channels required
6.2.1.5. Linkstochannels ratio
As the nodedegree constraint increases so does the number of links emanating from a mesh node. Since the links emanating from a node must be assigned different channels, this increases the NCR as well as decreasing the LCR for both CG approaches, as shown in Figures 1 and 2. However, the TCAbased approach still outperforms the MPbased approach by controlling the network connectivity using topology control, as shown next.
Since the solver finds a suboptimum solution of the multipath routing problem for a nodedegree constraint of 2 with the TCAbased approach for many of the CRTs, we move from Select 3 for less than 3 TCA to a higher TCA to build the CG until an optimum solution is found. This is the reason for the higher connectivity and interference in Figures 3 and 4 for a nodedegree constraint of 2 with the TCAbased approach.
6.2.2. Multipath vs. MPSP routing approach
We compare the performances of the multipath vs. MPSP routing approaches for different nodedegree constraints. For each routing approach, we use the TCAbased CG as well as the MPbased CG in combination with the MaISbased heuristic CA approach. For both routing approaches, we ask CPLEX to search for the first feasible solution, which significantly reduces the solution time at the cost of a small degradation in the network throughput, and set mipemphasis = 1 in CPLEX. For the TCAbased CG approach, we use the Select 3 for less than 3 TCA for the nodedegree constraint of 2 and 3, Select 4 for less than 4 TCA for the nodedegree constraint of 4, and so on. We are only interested in the first feasible solution for this comparison, so we do not move to a higher TCA in search of the optimum, unlike the previous comparison.
The MPbased CG performs poorly as compared to the TCAbased CG for both routing approaches in terms of solution time of the routing stage, NCR, and LCR and is slightly better than TCAbased CG in terms of network throughput. Therefore, for the sake of brevity, we present the results for each routing approach with TCAbased CG only in combination with the MaISbased CA heuristic.
6.2.2.1. Fairness
Total flow (multipath routing vs. MPSP routing (usingTCAbased CG))
Nodedegree constraint  Routing approach  y  Total flow  95% CI for total flow 

2  Multipath  1.160  40.59  37.9543.23 
MPSP  1.199  41.96  40.4043.53  
3  Multipath  1.902  66.56  62.1570.97 
MPSP  1.314  46.00  42.1949.82  
4  Multipath  2.382  83.37  78.1988.55 
MPSP  1.487  52.03  47.5556.52  
5  Multipath  2.981  104.33  97.01111.66 
MPSP  1.603  56.11  51.3360.89  
6  Multipath  3.593  125.74  116.67134.81 
MPSP  1.724  60.35  54.9965.71 
6.2.2.2. Network throughput
6.2.2.3. Solution time of routing stage
Solution time (multipath routing vs. MPSP routing (usingTCAbased CG))
Nodedegree constraint  Routing approach  Solution time (s)  95% CI for solution time 

2  Multipath  13.79  8.6018.98 
MPSP  1.64  1.471.82  
3  Multipath  1.68  1.481.88 
MPSP  0.25  0.230.27  
4  Multipath  1.75  1.432.08 
MPSP  0.24  0.240.25  
5  Multipath  1.89  1.372.42 
MPSP  0.26  0.260.27  
6  Multipath  2.37  1.693.05 
MPSP  0.28  0.280.29 
6.2.2.4. Number of channels required
6.2.2.5. Linkstochannels ratio
Since the objective of multipath routing is to maximize the total flow in the network by using multiple paths between the mesh nodes and the GW, it returns solutions in which more nodes have high nodedegrees, which means more links (conflicts) in the conflict graph. This leads to a higher NCR and a lower LCR as compared to MPSP routing. Although MPSP routing has smaller solution times, NCR, and LCR, it performs poorly vs. multipath routing in terms of network throughput at higher nodedegree constraints. There is a tradeoff between the network throughput and the NCR for the two routing approaches. For example, using the TCAbased CG and a nodedegree constraint of 6, multipath routing provides 2.08 times more network throughput than MPSP routing but is 1.67 times costlier than MPSP routing in terms of NCR.
6.2.3. MaISbased vs. MISbased heuristic approach for channel assignment
We compare the performances of MaISbased vs. MISbased heuristic approaches for CA for different nodedegree constraints based on NCR and LCR. For each CA approach, we use the two routing methods in combination with the two CG approaches. During routing, we search for the first feasible solution and use mipemphasis = 1 in CPLEX to reduce the solution time.
Since MPbased CG mostly performs poorly as compared to the TCAbased CG for both CA methods, we present the results for each CA method with the TCAbased CG only in combination with the two routing approaches.
6.2.3.1. Number of channels required
6.2.3.2. Linkstochannels ratio
The MaISbased heuristic not only outperforms the MISbased heuristic in terms of NCR and LCR but is also much more efficient in terms of computational complexity. The MaISbased heuristics return fewer channels than the MISbased heuristics because the cardinalities of the MaISs are more even in size whereas the MISbased heuristics first find a few large cardinality MISs, leaving a larger number of small MISs, for an overall larger number of channels.
7. Conclusions
We studied the problem of determining the minimum number of nonoverlapping frequency channels required by the mesh network to achieve maximum network throughput while maintaining fairness. We used our existing Select x for less than x TCA to build the CG with the objective of enhancing the spatial channel reuse in order to minimize the number of channels required. We found that the TCAbased CG approach outperforms the classical approach of MPbased CG in terms of solution time of the routing stage, NCR, and LCR for all nodedegree constraints. The TCAbased approach controls the network connectivity by controlling the neighborhoods of the mesh nodes, which leads to less overall transmitted power and better spatial channel reuse. We used multipath routing to achieve the maximum network throughput. This significantly outperforms the classical approach of MPSP routing in terms of network throughput at higher nodedegree constraints. However, we observed a tradeoff between network throughput and NCR for the two routing approaches. With an increasing nodedegree constraint, multipath routing provides more network throughput than MPSP routing but becomes costlier than MPSP routing in terms of NCR at the same time. We used heuristic algorithms to determine the minimum number of channels required for interferencefree CA, with the MaISbased heuristic CA outperforming the MISbased heuristic CA in terms of NCR and LCR for all nodedegree constraints.
We plan to extend our work in future using the signaltointerference ratiobased interference model for building the conflict graph. Further research will also focus on finding ways to significantly reduce the number of nonoverlapping frequency channels required.
Endnote
^{a}Preliminary work in this regard has been presented in ICCCN' 2013.
Declarations
Authors’ Affiliations
References
 Chaudhry AU, Ahmad N, Hafez RHM: Improving throughput and fairness by improved channel assignment using topology control based on power control for multiradio multichannel wireless mesh networks. EURASIP J. Wirel. Commun. Netw. 2012., 155(2012):Google Scholar
 Chaudhry AU, Hafez RHM, AboulMagd O, Mahmoud SA: Throughput improvement in multiradio multichannel 802.11abased wireless mesh networks. Miami: IEEE Globecom 2010; 2010:15. doi: 10.1109/GLOCOM.2010.5684193Google Scholar
 Raniwala A, Gopalan K, Chiueh T: Centralized channel assignment and routing algorithms for multichannel wireless mesh networks. ACM MC2R 2004, 8(2):5065.Google Scholar
 Kyasanur P, Vaidya N: Routing and interface assignment in multichannel multiinterface wireless networks. New Orleans: IEEE WCNC 2005; 2005:20512056. doi: 10.1109/WCNC.2005.1424834Google Scholar
 Ramachandran K, Belding E, Almeroth K, Buddhikot M: Interferenceaware channel assignment in multiradio wireless mesh networks. Barcelona: IEEE INFOCOM 2006; 2006:112.Google Scholar
 Avallone S, Akyildiz IF: A channel assignment algorithm for multiradio wireless mesh networks. Comput. Commun. 2008, 31(7):13431353. 10.1016/j.comcom.2008.01.031View ArticleGoogle Scholar
 Subramanian AP, Gupta H, Das SR, Jing C: Minimum interference channel assignment in multiradio wireless mesh networks. IEEE Tran. Mob. Comput. 2008, 7(12):14591473.View ArticleGoogle Scholar
 Karp RM Complexity of computer computations. In Reducibility among combinatorial problems. New York: Plenum Press; 1972:85103.Google Scholar
 Tang J, Xue G, Zhang W: Crosslayer design for endtoend throughput and fairness enhancement in multichannel wireless mesh networks. IEEE Trans. Wireless Commun. 2007, 6(10):34823486.View ArticleGoogle Scholar
 Rad AHM, Wong VWS: Crosslayer fair bandwidth sharing for multichannel wireless mesh networks. IEEE Trans. Wireless Commun. 2008, 7(9):34363445.View ArticleGoogle Scholar
 Giannoulis A, Salonidis T, Knightly E: Congestion control and channel assignment in multiradio wireless mesh networks. San Francisco: IEEE SECON 2008; 2008:350358. doi: 10.1109/SAHCN.2008.50Google Scholar
 Li XY, Nusairat A, Wu Y, Qi Y, Zhao JZ, Chu X, Liu Y: Joint throughput optimization for wireless mesh networks. IEEE Trans. Mob. Comput. 2009, 8(7):895909.View ArticleGoogle Scholar
 Ramamurthi V, Reaz A, Ghosal D, Dixit S, Mukherjee B: Channel, capacity, and flow assignment in wireless mesh networks. Comput. Netw. 2011, 55: 22412258. 10.1016/j.comnet.2011.03.007View ArticleGoogle Scholar
 Gardellin V, Das SK, Lenzini L, Cicconetti C, Mingozzi E, Pamela G: A divideandconquer approach for joint channel assignment and routing in multiradio multichannel wireless mesh networks. J. Parallel Distrib. Comput. 2011, 71: 381396. 10.1016/j.jpdc.2010.10.008View ArticleGoogle Scholar
 Brar G, Blough D, Santi P: Computationally efficient scheduling with the physical interference model for throughput improvement in wireless mesh networks. Los Angeles: ACM Mobicom 2006; 2006:213. doi: 10.1145/1161089.1161092Google Scholar
 Sen A, Murthy S, Ganguly S, Bhatnagar S: An interferenceaware channel assignment scheme for wireless mesh networks. Glasgow: IEEE ICC 2007; 2007:34713476. doi: 10.1109/ICC.2007.574Google Scholar
 Merlin S, Vaidya N, Zorzi M: Resource allocation in multiradio multichannel multihop wireless networks. Phoenix: IEEE INFOCOM 2008; 2008:610618. doi: 10.1109/INFOCOM.2008.110Google Scholar
 Li H, Cheng Y, Zhou C, Wan P: Multidimensional conflict graph based computing for optimal capacity in mrmc wireless networks. Genoa: IEEE ICDCS 2010; 2010:774783. doi: 10.1109/ICDCS.2010.58Google Scholar
 Wan PJ, Cheng Y, Wang Z, Yao F: Multiflows in multichannel multiradio multihop wireless networks. Shanghai: IEEE INFOCOM 2011; 2011:846854. doi: 10.1109/INFCOM.2011.5935308Google Scholar
 Fourer R, Gay DM, Kernighan BW: AMPL: a modeling language for mathematical programming. 2nd edition. Belmont, CA: Brooks/Cole; 2003.Google Scholar
 IBM ILOG CPLEX. . Accessed 05 February 2012 http://www01.ibm.com/software/integration/optimization/cplexoptimizer/
 Rappaport T: Wireless communications: principles and practice. 2nd edition. Upper Saddle River, NJ: Prentice Hall; 2002.Google Scholar
 Gupta P, Kumar PR: The capacity of wireless networks. IEEE Trans. Inf. Theory 2000, 46(2):388404. 10.1109/18.825799MathSciNetView ArticleGoogle Scholar
 Garey MR, Johnson DS, Stockmeyer L: Some simplified NPcomplete graph problems. Theor Comput Sc 1976, 1: 237267. 10.1016/03043975(76)900591MathSciNetView ArticleGoogle Scholar
 Wilf HS: Algorithms and complexity. London: Prentice Hall Inc; 1986.Google Scholar
 Jain R: The Art of computer systems performance analysis: techniques for experimental design, measurement, simulation, and modeling. New York: WileyInterscience; 1991.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.