Different Approximation Algorithms for Channel Scheduling in Wireless Networks

School of Computer Science, Wuhan University, Wuhan 430072, China Collaborative Innovation Center of Geospatial Technology, Wuhan 430072, China Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA LATNA, Higher School of Economics, Nizhni Novgorod, Russia Department of Computer Science, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA


Introduction
An important problem in wireless network is to develop efficient algorithms for maximum throughput by scheduling channels among many nodes. Based on the Shannon capacity formula, we can know that there is a minimum SINR required for each user in the network of wireless communication, as the existence of signal propagation loss during the process of data transmission; when the distance of a communication user pair is larger than the threshold, the effect of the received power in the receiver can be neglected because the receiver's SINR is less than the threshold. ere is an interference between two users in the same channel when their distance is less than a certain distance. Given a set of users, we find a way which schedules a maximal number of them without introducing interference between any two of them. e weighted version of this problem is to achieve the sum of maximal weights for the users assigned channels for communication. e channel scheduling problem is NPhard even in the single-channel model [1] and has been studied in a series of papers (for example, [2][3][4]). According to IEEE standard 802.11a, there are 13 orthogonal channels provided for wireless network [5]. e multichannel or multi-radio scheduling problem is essential to guarantee the performance of wireless networks.
A channel scheduling problem for a given graph G(V, E) is to select an edge subset E′ ∈ E and assign a channel to each edge in E ′ under the restriction so that all edges in E ′ are interference-free. e node exclusive interference model has been studied in many articles (for example, [6][7][8][9][10][11][12]). e algorithms for the single-channel or single-radio scheduling are reported in [13,14]. For the model not allowing two communication edges to share the same node, the optimal scheduling problem is converted into matching problem [15][16][17] which has a polynomial time solution. Some approximation algorithms for single channel were reported in [18] which show the existence of a theoretical polynomial time approximation scheme for the single-channel assignment. A simple greedy algorithm which is easy to implement is shown to have a constant factor approximation [18] for the single-channel assignment. e theoretical research about the multichannel scheduling has been reported in [19][20][21][22][23]. In [4], they formulate the channel allocation problem as an NP-hard nonlinear integer programming problem and then propose a probabilistic polynomial time (1 − (1/e))-approximation algorithm based on linear programming. A joint approach between routing and channel scheduling is shown in [24]. Some constant factor approximation algorithms are shown in [25] for assigning a minimum number of channels for a conflict-free communication, which are also NP-hard [26,27].
A polynomial time approximation scheme was developed in [28] for multi-radio and multichannel wireless network with very high computational complexity |E|O (d/∈ 2 ) for (1 − ϵ)-approximation, where d is the number of channels.
is theoretical result is not practical for implementation.
A maximal independent set problem based on joint scheduling and routing optimization is studied in [2], which is a mixed integer non-linear programming NP-hard problem. It developed a column generation based ϵ-approximation-bounded approximation algorithm, which can find tight ϵ-bounded approximate solutions and the optimal solutions.
We introduce the notion of two-side approximation for the channel scheduling problem. A pair of parameters (f, g) controls the accuracy of approximation in this paper. A (f, g)-approximation satisfies e∈E′ W(e) ≥ e∈Opt (W(e)/f) and e∈E− E′ W(e) ≤ g e∈E− Opt * W(e), where Opt * is the set of edges assigned channels in an optimal solution and W(e) is the weight of edge e. A f-approximation satisfies e∈E′ W(e) ≥ ( e∈Opt W(e)/f). Our two-side approximation approach combines two complementary problems. e first one is to maximize the total weights of communication edges that have been assigned channels, and the second one is to minimize the total weight of edges that do not receive channel for communication. It brings a more accurate tool for the approximation algorithms for the channel scheduling problem.
As the approximation scheme developed by Cheng et al. [28] has very high computational complexity, it is impossible to implement with software. Finding simple and efficient algorithm for the multichannel scheduling problem is still a challenging research topic. We show that a simple greedy algorithm can obtain an (O(1), O(1))-approximation for the single-channel scheduling problem. In many cases, the greedy algorithms give much more accurate results than the worst ratio. Furthermore, we develop an |E|O (1/ϵ) time (1 − ϵ, O(1))-approximation algorithm for the singlechannel scheduling problem. We also show that a simple greedy algorithm satisfies e∈E′ W(e) ≥ ( e∈Opt W(e)/Ω(1)), which can obtain an O(1)-approximation for the singlechannel scheduling problem. We also develop a |E|O (1/ϵ) time (1 − ϵ)-approximation algorithm for the multi-channel scheduling problem.
Our proposed multi-channel scheduling algorithm can be used in many wireless systems in which the number of users is larger than the number of subchannels; this case is considered in [29] where they study the channel assignment problem in uplink wireless communication system. It also can be used in the distributed data transmission system with limited communication resources, which is shown in [30]. e proposed methods are also suitable for channel assignment problem in the cellular-VANET heterogeneous wireless networks, which is studied in [25]. Our proposed channel scheduling approach has a wide range of applications and can be applied to different wireless resource allocation scenarios.
We also develop a polynomial time constant factor greedy approximation algorithm for the multi-channel scheduling that allows variate interference radius among those nodes. e paper is organized as follows: in Section 2, we present approximation algorithms for the weighted scheduling problems. e interference range is determined by a uniform parameter t, but the edges of communication have different weights. In Section 2.2, we show that a greedy algorithm for multi-channel assignment has a constant factor approximation. In Section 2.3, we show a two-side approximation algorithm for the single-channel assignment problem and show that there does not exist such an approximation for the multi-channel assignment problem. In Section 2.4, we give improved approximation schemes for both single-channel and multi-channel assignment problems. In Section 3, we develop the approximation for the unweighted channel assignment problem. is is the case that all the edges have weight 1, but the interference radius is not fixed. Some simulation results for the greedy approximation algorithm for the multi-channel assignment are given in Section 4. We draw conclusion in Section 5.

Weighted Channel Assignment
We develop approximation algorithms for the weighted channel assignment. e interference range is controlled by a fixed parameter t. e polynomial time approximation scheme for multi-channel scheduling problem is based on this model [28].

Definitions and Models.
e network is modeled as an edge-weighted graph G(V, E), where V is the set of nodes and E is the set of edges for traffic flow. e weight of each edge is often represented by the rate of network traffic. For each e ∈ E, let W(e) be the weight of e. Definition 1. Given an edge-weighted network graph G(V, E), let d(u, v) denote the distance between nodes u and v. e function d (u, v) can be either the geometric distance between u and v in Euclidean space if V is a set of points in Euclidean space or the hop distance between u and v in the graph G. e edge distance between two edges e 1 � (u 1 , u 2 ) and . Note that the graph G(V, E) only considers those edges E that need to do communication over a wireless network. No silent edge is included in E.
A set E ′ of edges is t-interference-free matching if E D(e 1 , e 2 ) > t for any two edges e and e ′ in E ′ , where t is distance threshold for interference.
A (multiple) channel assignment problem has a demand graph G � (V, E) that requests communication for (u, v) ∈ E. A wireless network may not have the resource to satisfy the communications for all edges. A channel assignment algorithm selects a subset E ′ ⊆ E and assigns a channel to each edge in E ′ so that the edges in the same channel form a t-interference-free matching, where t is distance threshold for interference. When there is only one channel available for the entire demand graph, the channel assignment problem is called single-channel assignment problem. Otherwise, it is called multi-channel assignment problem.
A channel assignment for an edge e � (u, v) is represented by (e, K), where K is a channel. Assume that M is a set of channel assignments. A channel assignment (e, K) is interfered by M if there is a channel assignment (e′, K) ∈ M such that E D(e, e ′ ) ≤ t. Define M * to be the set of all edges e with (e, K) ∈ M for some channel K. If W(.) is the weight function for the edges in G, define W(M * ) � e∈M * W(e). An optimal solution for a channel assignment problem G � (V, E) with weight function W(.) is a set Opt of channel assignments for edges such that W(Opt * ) is the maximum.

Definition 2.
Assume that A is a set of channel assignments for G.
where τ A (e) is the interference degree of an edge e ∈ E, τ A (G) is the interference degree of the graph G � (V, E), and t is distance threshold for interference.
If the distance is the Euclidean distance and all nodes are on the plane, then τ(G) ≤ 11, which is shown in Lemma 1. If the distance is the hop distance and all nodes are on the plane, then τ(G) ≤ 49, which was proved in [18]. erefore, τ(G) does not depend on the threshold.

Definition 3.
Assume that G is a channel scheduling problem. We define some measures for approximations.
(i) An f-IS-approximation App for the channel scheduling problem satisfies the condition W(App * ) ≥ (W(Opt * )/f). (ii) A g-VC-approximation App for the channel scheduling problem satisfies the condition (iii) A (f, g)-approximation for the channel scheduling problem satisfies the conditions W(App * ) ≥ (W(Opt * )/f) and W(E − App * ) ≤ gW(E − Opt * ).

Lemma 1.
Assume that all nodes in the demand graph G � (V, E) are points on a plane. en, τ(G) ≤ 11 for the Euclidean distance t as the threshold for the interference.
Proof. Let e � (u, v) be an edge in the graph G. e distance between u and v is at most t (otherwise, they cannot communicate). Edge e and another edge e ′ � (u ′ , v ′ ) have interference if one of u and v and one of u ′ and v ′ have distance at most t. Let each node be a center of a circle of diameter t/2 on a plane. Let C(p, r) represent a circle with center at p and radius r. Let A be a channel assignment for G, which gives a list of disjoint circles C 1 , · · ·, C m such that each node in those edges with channel assigned is a circle center. Circle C(u, t/2) touches at most 6 circles among C 1 , · · ·, C m , and so does C(v, t/2). If both C(u, t/2) and C(v, t/2) touch 6 circles, there must exist at least one of them touched by both C(u, t/2) and C(v, t/2) because the distance between u and v is at most t.

Multiple Channel with Fixed Interference Radius
Scheduling. In this section, we present a greedy approximation algorithm for the multi-channel scheduling problem shown in Algorithm 1. We can only show a one-side O(1)-approximation for the multi-channel scheduling problem.
In the multi-channel scheduling problem, we assume that each node has d channels available for allocation. Two nodes u and v can communicate if their corresponding edge e � (u, v) is assigned a channel (the distance between u and v is at most the threshold t) and has no interference with other edges in the same channel.
We note that when there is a set F of edges that have been assigned channels, it is straightforward to check if a new edge e will be interfered at certain channel K. is can be done by checking E D(e, e i ) ≤ t for all e i ∈ F K , where F K is the subset of edges in F assigned channel K.
For the case that all nodes are on a plane, and the distance is either Euclidean distance or hop distance, the following theorem gives a constant factor approximation for the multiple channel assignment problem as τ(G) is bounded by constants in both distances.

Theorem 1.
e Algorithm 1 greedy-multi-channel (.) is a g(α)-IS-approximation algorithm for the multi-radio multichannel scheduling problem and has the computational complexity Proof. Assume that Opt is an optimal solution for the channel scheduling problem. Let App be an approximate solution derived by greedy-multi-channel (.). For each edge g in Opt * , assign an edge e, denoted by H(g), in App * such that R(e) is the least among all edges with interference with g. When the algorithm greedy-multi-Mobile Information Systems 3 channel (.) processes an edge e � (u, v), the available channels for e are those which are not allocated to the edges with end points at either u or v.
Proof. It follows from the greedy algorithm which processes the edges according the decreasing order of their weights. Let d be the total number of channels. For each edge e � (u, v) ∈ Opt * − App * , we consider two cases. e constant α used in the two cases will be assigned later.

Case 1.
ere are at least α d channels available for the edge e when e is processed. Such an edge e is of type 1.
In this case, since e is not assigned for a channel, there must be at least α d edges e ′ that are already assigned channels and have E D(e, e ′ ) ≤ t. For such an edge e ′ with E D(e, e ′ ) ≤ t, let b(e, e ′ ) � W(e)/α d. Since e ′ is processed before e in the algorithm, we have W(e) ≤ W(e ′ ). (1) For each edge e ′ in App * , we have ere are fewer than α d channels available for the edge e � (u, v) when e is processed in greedy-multi-channel (.). Such an edge e is of type 2.
Each edge in Opt * − App * is of either type 1 or type 2. We have (1) and (1) (1) and (1)) We have (1)) Input: a weighted graph G, and a distance parameter t for the largest distance of interference. Output: L′ (1) Sort all edges by the increasing order of their weights and put those edges in a list L; (2) Let L′ � ∅; (3) Repeat; (4) Select an edge e � (u, v) with the largest weight from L; (5) If there is a channel K for e not being interfered by L′ (6) en, assign the channel K to e and put (e, K) into L′; (7) Remove e from L; (8) Until L is empty; (9) Return L′; ALGORITHM 1: Greedy-multi-channel with fixed interference radius. 4 Mobile Information Systems . us, the ratio of approximation is g(α).

Single Channel with Fixed Interference Radius Scheduling.
In this section, we present a greedy (O(1), O(1)) approximation algorithm for the channel scheduling problem shown in Algorithm 2. Our two-side approximation bound for the greedy algorithm improves the one-side approximation bound in [18].
We note that when there is a set F of edges that have been assigned channels, it is straightforward to check if a new edge e will be interfered at certain channel K. is can be done by checking E D(e, e i ) ≤ t for all e i ∈ F K , where F K is the subset of edges in F in channel K.
Proof. Assume that Opt is an optimal solution for the channel scheduling problem. Let App is an approximate solution derived by greedy-single-channel (.).
For each edge g in Opt * , assign a edge e, denoted by H(g), in App * such that e has the largest weight among all edges with interference with g.
Consider an edge e � (u, v) selected in App * . Let A(e) � e 1 , · · ·, e m be the list of edges with H(e i ) � e. Claim 2. For each edge g in Opt, W(g) ≤ W(H(g)).
Proof. If g is in App * , we consider an edge has interference with itself. us, it is trivial. By the definition of H(g), H(g) has the largest of weight among all edges in Opt * with interference with g.
Assume that g ′ is the first edge in App * and has the same channel with g ∈ Opt * . Before selecting g ′ for assigning a channel, there is no interference between g and other edges in App. erefore, W(g ′ ) ≥ W(g) (otherwise, g ′ should not be selected for channel assignment). Since H(g) has the largest weight among all edges with interference with g in Assume that App contains channel assignments for edges e 1 , · · ·, e m . Partition Opt * into A(e 1 ), · · · , A(e m ). By Claim 2, we have that erefore, we have τ(G)W(App * ) ≥ W(Opt * ). us, App is a τ(G)-IS approximation for G.
On the other hand, for each e i ∈ (E − Opt * ) ∩ App * , we always have τ(e i )W(e i ) ≥ e′∈A(e i ) W(e ′ ) (by Claim 2). For each edge e ′ in (E − App * ) ∩ Opt * , there is an edge e ∈ App * such that e and e ′ have the interference at the same channel (otherwise, e ′ would be assigned some channel by Greed1 (.)). Furthermore, e ∉ Opt * since e has interference with e ′ . erefore, there is an edge e * ∈ (E − Opt * ) ∩ App * such that e ′ ∈ A(e * ). By inequality (1), we have We have the inequalities: erefore, greedy-single-channel (.) gives a (τ(G), τ(G))-approximation. Proof. For the implementation, we partition the plane into grid of size t × t. When an edge is assigned a channel, put the assignment into the corresponding grid for one of its two nodes in the edge. When assigning a channel a new edge, check the assigned edges in the nearby O(1) grids. us, each edge only costs an O(1) time.
If d(u, v) is defined to be the hop distance, then τ(G) is at most 49, which is shown in [18]. If d(u, v) is the Euclidean distance, we show that τ(G) is at most 11. Proof. It follows from eorem 2 and the fact τ(G) ≤ 49, which is shown in [18]. A brute force implementation takes O(|E| 2 ) time.
For the multi-channel scheduling problem, we show that it does not have two-side approximation unless P�NP.
it becomes an optimal solution. us, we can search the least number of channels from 1 to k to support all edges in G. is brings a polynomial time solution for the conflict-free channel assignment problem, which was proved to be NP-hard [26]. erefore, P�NP.
We have the following corollary that shows we cannot have a two-side approximation for the multi-channel scheduling problem.

Corollary 1. Assume that f(n) is a function from
en, there is no polynomial time (f(n), f(n))-approximation for the multiple channel scheduling problem unless P�NP.
In Section 2.4, we show a faster approximation for (1 − ϵ)-IS-approximation than that in [28]. Part of the algorithm is based on the shifting technology which has originated from [31] and has been widely used in developing approximation for networking problem (for example, [18,28]).
In Section 3.1, we present a greedy O(1)-approximation algorithm for the unweighted multi-channel scheduling problem with variate interference radii.
In Section 3.2, we present a greedy (O(1), O(1)) approximation algorithm for the unweighted single channel with variate interference radii scheduling problem.
In Section 4, we did some simulation for the multichannel assignment problems for the greedy algorithm. Greedy algorithm is easy to implement and fast to output the result. Its experimental results show much better performance than the theoretical approximation ratio, which is derived under the worst case analysis.

Improving the Existing Algorithm for the Multichannel
Scheduling. We show a faster approximation for (1 − ϵ)-ISapproximation than that in [28]. Let PTAS (.) (polynomialtime approximation scheme) represent the algorithm described below. Part of the algorithm is based on the shifting technology which has originated from [31] and has been widely used in developing approximation for networking problem (for example, [18,28]).
Assume that the maximal distance of two nodes for communication is one. Let t be the distance of interference. e grid size is D � t + 2. e plane is partitioned into grids of size D × D.
Let ϵ be a constant in (0, 1) Theorem 6. Assume that ϵ is an arbitrary constant in (0, 1). en, there is an |E| O(d/ϵ) time algorithm to give a (1 − ϵ)-ISapproximation for the multi-channel scheduling with d channels.
Require: a weighted graph G, and a distance parameter t for the largest distance of interference. Ensure: L′; (1) Sort all edges by the decreasing order of their weights and put those edges in a list L; (2) Let L′ � ∅; (3) Repeat; (4) Select an edge e � (u, v) with the largest weight from L; (5) If e is not being interfered by L′; (6) en, assign the channel K to e and put (e, K) into L′; (7) Remove e from L; (8) Until L is empty; (9) Return L′; ALGORITHM 2: Greedy-single-channel with fixed interference radius (G, t). 6 Mobile Information Systems

Proof. A point is in the boundary of G i,j if it is in a D × D grid in the boundary of a m D × m D big grid of P i,j . Let E B,O i,j
be the set of all edges that connect the points in the boundary of G i,j in an optimal solution. Assume P i,j is the disjoint union of m D × m D grids Q 1 ∪ Q 2 ∪ , . . . , Q t . Let O(Q i ) be the optimal solution for the set of edges connecting at least one node not in any D × D boundary grid of Each boundary grid is in at most (2m − 1) P i,j . We also have erefore, there are i 0 and j 0 such that By inequalities (11) and (12), we have

Merging Two Algorithms in Single-Channel Scheduling.
In this section, we show that merging Algorithm 1 and PTAS (.) can speed up the polynomial time approximation scheme in many cases for the single-channel scheduling. We propose Algorithm 3 which shows a polynomial time (1 − ϵ, O(1))-approximation scheme for the single-channel scheduling problem. (1 − g(1 − c))-IS-approximation.

Lemma 3. Assume that a channel scheduling problem G satisfies the condition W(Opt) ≥ cW(E). en, a g-VC-approximation for the channel scheduling implies a
Proof. Assume an approximation App satisfies Theorem 7. Assume that ϵ is an arbitrary constant in (0, 1).

en, there is an |E| O(1/ϵ) time algorithm to give a (1 − ϵ, O(1))-approximation for the single-channel scheduling. Furthermore, Algorithm 3 runs in O(|E|
Proof. By eorem 2, we have that App 1 is a (τ(G), τ(G))-approximation for the single-channel sched- Since App 2 is a (1 − ϵ)-IS-approximation for the channel scheduling problem and App 2 is also an approximation for the channeling problem, we have W( ) being at most W(Opt * ) and at least W(Opt * )/(1 − ϵ). erefore, App i is a (1 − ϵ)-IS-approximation for the channel scheduling problem.

Unweighted Channel Assignment
In this section, we develop the approximations for the unweighted channel assignment problem. is is the case that all the edges have weight 1. e radii of interference are not fixed.
In the unweighted model, each node u has a radius r(u) for the range that u has the interference. Any node v with distance at most r(u) to u is interfered by u when u is active. To an edge e � (u, v) in an active communication, we assume dist(u, v) ≤ min(r(u), r(v)).
For edges e � (u 1 , u 2 ) and e ′ � (v 1 , v i ), there is an interference between them if dist(u i , v j ) ≤ max(r(u i ), r(v j )) for some u i and v j in the nodes of the two edges.

Definition 4.
Assume that G is a channel scheduling problem. We define some measures for approximations.
An optimal solution for G is a set Opt * of edges without interference with largest |Opt * |. A f-IS-approximation App for the channel scheduling problem satisfies the condition |App * | ≥ (|Opt * |/f). A g-VC-approximation App for the channel scheduling problem satisfies the condition |E − App * | ≤ g|E − Opt * |. A (f, g)-approximation for the channel scheduling problem satisfies the conditions |App * | ≥ (|Opt * |/f) and |E − App * | ≤ g|E − Opt * |.
By Definition 3, a (f, g)-approximation for the channel scheduling problem is both f-IS-approximation and g-VCapproximation for it. Many existing papers used the f-ISapproximation to measure the accuracy.

Mobile Information Systems 7
Definition 5. Assume that A is a set of channel assignments for G.
there is an interference between e and e ′ }|. Proof. Let e � (u, v) be an edge in the graph G. We only consider a fixed channel K. Let C(p, r) represent a circle with center at p and radius r. Define I(u) to be the set of edges e ′ � (u ′ , v ′ ) ∈ A assigned with channel K such that u has interference with u ′ or v ′ (in other words, dist(u, u ′ ) ≤ min(r(u), r(u ′ )) or dist(u, v ′ ) ≤ min(r(u), x is a node in e ′ ∈ I(u) with r(u) > r(x)}.
Assume that d * A (G) is large. Consider the case that node u has interference with other edges. For the case of v, we have similar conclusions.
Using u as the center, evenly partition the plane into the h fan areas with angle (2π/h) each. By the pigeon hole principle, there is a fan area F i that has at least (d * A (G)/4h) ≥ 2. In this case, the node x L with the largest radius interferes all other nodes of F i . is is a contradiction since all nodes F i are from the solution A and have no interference each other. J ≤ (u) ≤ 24.

Multiple Channels with Variate Interference Radii.
In this section, we present a greedy O(1)-approximation algorithm for the unweighted multi-channel scheduling problem with variate interference radii, which is shown in Algorithm 4.
We note that when there is a set F of edges that have been assigned channels, it is straightforward to check if a new edge e will be interfered at certain channel K. is can be done by checking E D(e, e i ) ≤ t for all e i ∈ F K , where F K is the subset of edges in F in channel K.

Theorem 8. e Algorithm 4 greedy (.) is a g(α)-IS-approximation algorithm for the multi-radio multi-channel scheduling problem and has the computational complexity
Proof. Assume that Opt is an optimal solution for the channel scheduling problem. Let App be an approximate solution derived by greedy (.). For each edge g in Opt * , assign an edge e, denoted by H(g), in App * such that R(e) is the least among all edges with interference with g.
Proof. If g is in App * , we consider an edge has interference with itself. us, it is trivial. By the definition of H(g), R(H(g)) is the least among all edges in Opt * with interference with g.
Assume that g ′ is the first edge in App * and has the same channel with g ∈ Opt * and has interference with g. Before selecting g ′ for assigning a channel, there is no interference between g and other edges in App. erefore, R(g ′ ) ≤ R(g) (otherwise, g ′ should not be selected for channel assignment). Since H(g) has the least R (H(g)) among all edges with interference with g in App, we have R(H(g)) ≤ R(g ′ ) ≤ R(g).
Let d be the total number of channels. For each edge e � (u, v) ∈ (Opt * − App * ), we consider two cases.

Case 1.
ere are at least α d channels available for the edge e when e is processed. Each of such an edge is called type A.
In this case, since e is not assigned for a channel, e must have interference with at least α d edges already assigned channels. For each edge e ′ with interference with e, let Require: constant ϵ ∈ (0, 1) and graph G(V, E). Ensure: App i (1) Let App 1 � App 2 � ∅; (2) Run greedy-single-channel (.) to find an (O(1), O(1))-approximation App 1 ; We have We select the constant α so that . us, the ratio of approximation is g(α).

Single Channel with Variate Interference Radii.
In this section, we present a greedy (O(1), O(1)) approximation algorithm for the unweighted single channel with variate interference radii scheduling problem; the pseudocode is shown in Algorithm 5 as follows.
We note that when there is a set F of edges that have been assigned channels, it is straightforward to check if a new edge e will be interfered at certain channel K. is can be done by checking if e has interference with any e i ∈ F K , where F K is the subset of edges in F in channel K.

Theorem 9.
e Algorithm 5 greedy (.) is an (d * A (G), d * A (G))-approximation algorithm for the multi-radio multi-channel scheduling problem and has the computational complexity O(|E| 2 ).
Proof. Assume that Opt is an optimal solution for the channel scheduling problem. Let App be an approximate solution derived by greedy (.).
For each edge g in Opt * ∞, assign a edge e, denoted by H(g), in App * such that R(e) is the least among all edges with interference with g.
Consider  Proof. If g is in App * , we consider an edge has interference with itself.
us, it is trivial. By the definition of H(g), R(H(g)) is the least among all edges in Opt * with interference with g.
Assume g ′ is the first edge in App * and has the same channel with g ∈ Opt * and has interference with g. Before selecting g ′ for assigning a channel, there is no interference between g and other edges in App. erefore, R(g ′ ) ≤ R(g) (otherwise, g ′ should not be selected for channel assignment). Since H(g) has the least R (H(g)) among all edges Require: a weighted graph G, and a distance parameter t for the largest distance of interference. Ensure: L′ (1) Sort all edges by the increasing order of their sum of radii and put those edges in a list L.
with interference with g in App, we have Assume that App contains channel assignments for edges e 1 , . . . , e m . Partition Opt * into A(e 1 ), . . . , A(e m ). By Claim 1, we have that On the other hand, for each e i ∈ (E − Opt * ) ∩ App * , we always have d * A (e i ) ≥ |A(e i )| (by Claim 1). For each edge e ′ in (E − App * ) ∩ Opt * , there is an edge e ∈ App * such that e and e ′ have the interference at the same channel (otherwise, e ′ would be assigned some channel by Greed (.)). Furthermore, e ′ ∉ Opt * since it has interference with e. erefore, there is an edge e * ∈ (E − Opt * ) ∩ App * such that e ′ ∈ A(e * ). By inequality (1), we have We have the inequalities is gives |E − App * | ≤ d * A (G)|E − Opt * |. erefore, greedy (.) gives a (d * A (G), d * A (G))-approximation.

Experimental Results
Greedy algorithm is easy to implement and fast to output the result. We did some simulation for the multi-channel assignment problems for the greedy algorithm. e parameter T represents a distance, which is either Euclidean distance or hop distance. In a tuple (a, b), the first item a is the total weight of an optimal solution (W(Opt * )), and the second item b is the total weight of the approximation solution (W(App * )).
We also evaluate the effectiveness of the proposed greedy algorithm; the algorithm performance is compared with the baseline random assignment algorithm, the greedy coloring algorithm which is proposed in [32], and the conventional graph coloring algorithm (GC) in [33]. ese three methods are briefly described as follows: Random: we allocate channels randomly for each node. Greedy coloring: we allocate the best channel to each node greedily and then transfer it to robust graph coloring problem which aims at minimizing the interference of system. GC: we use an interference negligible distance to find which pairs of users can share channels with the cellular users.

Multichannel with Hop Distance.
Our experimental results for the hop distance show that the results from simulation are much better than the theoretical analysis for the greedy algorithm. In this experiment, we randomly generate a network with 30 nodes and 50 edges. For each edge, we generate a random number as a weight. Replacing with the optimal solution of total weight, we get the following simulation. e performance of the algorithm is shown in Figure 1. We note that the performance of the greedy algorithm goes down as the threshold of the distance parameter T goes up. We can also observe that the performance of our greedy algorithm is superior to the greedy coloring, GC, and random algorithms.
Input: a weighted graph G, and a distance parameter t for the largest distance of interference. Output: L′ (1) Sort all edges by the increasing order of their sum of radii and put those edges in a list L.
(2) Let L′ � ∅; (3) Repeat; (4) Select an edge e � (u, v) with the least sum of radii from L; (5) If there is a channel K for e not being interfered by L′ (6) en, assign the channel K to e and put (e, K) into L′; (7) Remove e from L; (8) Until L is empty; (9) Return L′; ALGORITHM 5: Greedy-signal-channel with variate interference radii (G, t).

Multichannel with Euclidean
Distance. Our experimental results for the Euclidean distance show that the results from simulation are much better than the theoretical analysis for the greedy algorithm. In this experiment, we randomly generate a network with 30 nodes and 50 edges in the space of 500 × 500. For each edge, we generate a random number as a weight. Replaced with the optimal solution of total weight, the following simulation has been achieved.  e performance of the algorithm is shown in Figure 2. From Figure 2, we can also know that the performance of the greedy algorithm degrades as the threshold of the distance parameter T increases from 10 to 100. e performance of the proposed greedy algorithm is better than the other three comparison algorithms.

Conclusion
We develop a new measure, which controls the ratios in two sides, for the approximation algorithms for channel scheduling problem. We first study the weighted multichannel and single-channel assignment with fixed interference range problems and propose corresponding algorithms for these two problems; furthermore, we improve the existing algorithms for the multi-channel scheduling and propose a faster and more accurate (1 − ϵ, O(1))-approximation algorithm for signal channel scheduling problem. Next, we study the unweighted channel assignment with variate interference radius problem and present a multichannel and a single-channel allocation algorithm, respectively, for this situation. All of the algorithms are given detail theoretical proofs and got constant approximation guarantees. Finally, we verified our algorithms with simulations.

Data Availability
e data used to support the findings of this study are included within the article.

Disclosure
An earlier version of this paper was presented at the "Wireless Algorithms, Systems, and Applications 6th International Conference WASA 2011 Chengdu China August 11-13 2011." We have added Section 2.4 "Improving the Existing Algorithm for the Multi-Channel Scheduling," Section 2.5 "Merging Two Algorithms in Single-Channel Scheduling," Section 3 "Unweighted Channel Assignment," and Section 4 "Experimental Results."

Conflicts of Interest
e authors declare that they have no conflicts of interest.    Algorithm for the Multi-Channel Scheduling," Section 2.5 "Merging Two Algorithms in Single-Channel Scheduling," Section 3 "Unweighted Channel Assignment," and Section 4 "Experimental Results."