A Column Generation Method for Spatial TDMA Scheduling in Ad Hoc Networks

Transcription

1 A Column Generation Method for Spatial TDMA Scheduling in Ad Hoc Networks Patrik Björklund, Peter Värbrand, Di Yuan Department of Science and Technology, Linköping Institute of Technology, SE , Norrköping, Sweden Abstract An ad hoc network can be set up by a number of units without the need of any permanent infrastructure. Two units establish a communication link, if the channel quality is sufficiently high. As not all pairs of units can establish direct links, the traffic between two units may have to be relayed through other units. This is known as the multi-hop functionality. In military command and control systems, ad hoc networks are also referred to as multi-hop radio networks. Spatial TDMA (STDMA) is a scheme for access control in ad hoc networks. STDMA improves TDMA by allowing simultaneous transmission of multiple units. In this paper, we study the problem of STDMA scheduling, where the objective is to find minimum-length schedules. Previous work for this problem has focused on heuristics, whose performance is difficult to analyze when optimal solutions are not known. We develop novel mathematical programming formulations for this problem, and present a column generation solution method. Our numerical experiments show that the method generates a very tight bound to the optimal schedule length, and thereby enables optimal or near-optimal solutions. The column generation method can be used to provide benchmarks when evaluating other scheduling algorithms. In particular, we use the bound obtained in the column generation method to evaluate a simple greedy algorithm that is suitable for distributed implementations. Key words: Ad hoc networks, STDMA, scheduling, column generation Corresponding author. address: patbj, petva, diyua@itn.liu.se (Patrik Björklund, Peter Värbrand, Di Yuan). Preprint submitted to Elsevier Science 19th June 2003

2 1 Introduction An ad hoc network consists of a collection of radio units with a wireless interface forming temporary connections. No fixed infrastructure is involved in the communication. Instead, two radio units establish a direct communication link, if the signal-to-noise ratio is high enough. Two radio units far away from each other may communicate, if the units between them are participating in the ad hoc network, and are willing to forward packets for them (so called the multi-hop functionality). This type of network is often referred to as multi-hop radio networks in military command and control systems, for which it is often not feasible to install any permanent communication infrastructure. In recent years, there is a growing interest in other applications of ad hoc networks, such as peer-to-peer computer communications, and communications between mobile sensors (e.g., traffic safety systems). As pointed out in [10], ad hoc networks pose many design challenges. In this paper, we address the issue of resource allocation when designing link access schemes. One access scheme for ad hoc networks is Time Division Multiple Access (TDMA), in which the transmission resource of a radio frequency is divided into time slots, and each unit (or each link) receives a dedicated slot. Although simple to implement, TDMA is very inefficient from the resource utilization point of view. A promising approach to increase the network capacity is Spatial TDMA, or STDMA [18], which takes into account the fact that radio units are usually spread out geographically, and hence units with a sufficient spatial separation can use the same time slot for transmission. In STDMA, the efficiency of the spatial reuse depends on the algorithm used for generating the transmission schedule. Scheduling algorithms become thus very important for implementing STDMA. A number of scheduling problems, with different levels of complexity, can be identified in this context. From an algorithmic perspective, the most challenging problem is to derive distributed algorithms that can handle mobile scenarios, and can perform STDMA scheduling based on the traffic distribution in the network. However, to design such algorithms, solutions to simple scenarios are very useful. The most simple version of the problem is to compute a schedule centrally for a static network, without taking the traffic distribution into account. In this case, the objective is to find a schedule that is as short as possible, i.e., a minimum-length schedule in which each unit (or each link) receives at least one time slot. In this paper, we will focus on this fundamental problem. Even if this is the most simple scenario of STDMA scheduling, finding the optimal solution to the problem is difficult. In particular, it can be shown that, from an computational complexity point of view, the problem is NP -hard. Previous work for this problem (e.g. [4], [7], [12], [15], and [19]) has focused on heuristics. However, in the absence of optimal solutions, it is hard to judge the performance 2

3 of heuristics, and to assess the true potential of STDMA. In this paper, we present novel linear integer formulations for minimum-length STDMA scheduling. Moreover, we present a column generation method that effectively exploits the structure of the formulations. The main strength of the methodology is its scalability. The method can be used to find optimal or nearoptimal schedules for networks with arbitrary topology and realistic size. To the best of our knowledge, the formulations and the solution methodology have not been applied to STDMA scheduling before. Using the solutions provided by the column generation method, we are able to evaluate a simple heuristic that is suitable for distributed implementation. The remainder of this paper is organized as follows. We present our network model in Section 2. In Section 3, two assignment strategies for STDMA scheduling are discussed. The optimization problems are formalized in Section 4, and the computational complexity is studied in Section 5. We present mathematical formulations in Section 6, and the column generation method in Section 7. In Section 8 we discuss two approaches to obtain feasible schedules. Numerical results are presented in Section 9. In Section 10, we draw some conclusions and outline future work. 2 The Network Model An ad hoc network can be characterized by a directed graph G = (N, A), where the node set N represents the radio units, and the arc set A represents the communication links. A directed link (i, j) is established if the signal-tonoise ratio (SNR) is greater or equal to a given threshold, that is, if SNR(i, j) = P i L b (i, j)n r γ 0, (1) where P i is the transmitting power of i, L b (i, j) is the path-loss between i and j, N r is the effect of the thermal noise, and γ 0 is the threshold. We assume that the transmitting power of the nodes are given constants (e.g., every node transmits at its maximum power when sending data to another node). Typically, the cardinality of A is much less than N ( N 1), meaning that the network is sparsely connected. A sample network of 20 nodes is shown in Figure 1. Remark It is often assumed (e.g., [13] and [14]) that P i = P, i N, and L b (i, j) = L b (j, i), (i, j) A. These assumptions are, however, not necessary for our mathematical models or solution methods. 3

4 Figure 1. An ad hoc network with 20 nodes. Several assumptions are commonly made in STDMA scheduling. First, a node cannot transmit and receive simultaneously. Secondly, a node can receive data from at most one other node at any time. Finally, we assume that a link is error-free only if the signal-to-interference ratio (SIR) is above a threshold γ 1 (possibly equals γ 0 ). For link (i, j), the SIR-criterion is formulated as SIR(i, j) = P i L b (i, j)(n r + k K,k i P k ) γ 1. (2) L b (k,j) In (2), K is the set of nodes that are in simultaneous transmission. The term is thus the accumulated interference with respect to link (i, j). k K,k i P k L b (k,j) 3 Assignment Strategies In STDMA, the access control at the link layer is implemented using a transmission schedule. The schedule consists of a number of time slots. One or several network units are permitted to transmit in each of the slots. The length of the schedule determines the size a data frame, and the schedule repeats itself from one frame to the next. There are two possibilities for assigning the time slots: node-oriented assignment and link-oriented assignment. In the former strategy, a node is assigned one or several slots in the schedule. In each of these time slots, the node may use any of its (outgoing) links for transmitting data to another node. This assignment strategy is well-suited for broadcast traffic. In link-oriented assignment, a link is assigned one or several time slots for point-to-point communication between a specific pair of nodes. Empirically, link-oriented assignment achieves a higher spatial reuse than node-oriented assignment (see [12]). 4

5 Note that the SIR-criterion (2) leads to different constraints in the two assignment strategies. For node-oriented assignment, a time slot can be assigned to node i only if all the outgoing links of i satisfy (2). If a time slot is assigned to link (i, j) using link-oriented assignment, then it is required that (2) is satisfied for this particular link. A variety of heuristic algorithms for STDMA scheduling can be found in the literature. Algorithms for node-oriented assignment are proposed in [2], [4], [5], and [7], while [3], [11], [15], and [19] focus on algorithms using link-oriented assignment. Comparisons of the two assignment strategies can be found in [12], [13], [14], and [21]. 4 Problem Definition If traffic distribution is not taken into consideration, then the length of the STDMA schedule determines the efficiency of the spatial reuse of the time slots. We define two optimization problems, denoted by MNP and MLP, for minimum-length scheduling for node-oriented and link-oriented assignments, respectively. Given the set of nodes N, the path-loss between every pair of nodes (i.e., L b (i, j), i, j N : i j), the transmitting power of each node (i.e., P i, i N), the noise effect N r, and the two threshold values γ 0 and γ 1, the objective of MNP is to find a minimum-length schedule, such that every node receives at least one time slot, and such that the following are satisfied. Two end nodes of a link must be assigned different time slots. (This is because a node cannot transmit and receive in a time slot.) Two nodes, both having directed links to a third node, must be assigned different time slots. (This is because a node cannot receive from more than one other node in a time slot.) A time slot can be assigned to a node only if all the outgoing links of the node satisfy the SIR-constraint (2). For link-oriented assignment, the corresponding problem MLP amounts to finding a minimum-length schedule, such that every link receives at least one time slot, and such that the following are satisfied. Two links that share a common node, irrespective of the link directions, must be assigned different time slots. (This constraint comprises the first two constraints of MNP.) A time slot can be assigned to a link only if the SIR-constraint (2) for the link is satisfied. 5

6 Figure 2. Graph construction in the N P -hardness proof. 5 Computational Complexity We show that, from the computational complexity point of view, both problems defined in the previous section are NP -hard. Proposition 1 Problem MNP is N P -hard. Proof Consider the graph coloring problem defined for an undirected graph G = (V, E). We construct, in polynomial time, an instance of MNP, such that the two problems are equivalent. For every edge (i, j) E, we define a node v ij. Let V E = {v ij, (i, j) E}. The set of nodes to MNP is then defined as N = V V E. The set of directed links A comprises two parts. First, for each node v ij V E, we define two directed links, (i, v ij ) and (j, v ij ). Secondly, for every pair of nodes in V E, v ij and v kl, we defined two directed links, (v ij, v kl ) and (v kl, v ij ). Figure 2 shows the graph construction for a small network of four nodes. We let γ 0 = γ 1 = 1, P i = 2, i N, N r = 1, and L b (i, j) = 1, (i, j) A. For all the pairs of nodes that do not have a link in A, we choose a sufficiently large path-loss value, such that the SNR-constraint for these node pairs are not satisfied, and, in addition, the SIR-constraints for links in A are redundant. (A particular path-loss value that satisfies these conditions is 2 N.) We show that the instance of MNP defined for the node set N and link set A, along which the other problem parameters defined as above, is equivalent to the original graph coloring problem. To see the equivalence, we make the following observations. First, the nodes in the set V E must be assigned different time slots (or colors). Secondly, the node sets V and V E must use disjoint sets of colors. Finally, two adjacent nodes in the graph coloring problem cannot use the same time slot. These observations lead directly to the conclusion that a feasible solution of one problem corresponds to a feasible solution of the other problem, and vice versa. For any of such pairs of solutions, the difference in the objective values is a constant (equals E ). We further note that the reduction 6

7 itself is polynomial. Hence the conclusion. Proposition 2 Problem MLP is N P -hard. Proof Consider the edge coloring problem defined for an undirected graph G = (V, E). We construct, in polynomial time, an instance of MLP, such that the two problems are equivalent. For each edge (i, j) E, we define a directed link from i to j. Denote the set of directed links by A. We further let N = V. We choose the values of transmitting power, the noise effect, the path-loss parameters, and the threshold values such that the SNR-criterion is satisfied for nodes i and j if and only if (i, j) A, and such that the SIR-constraints are redundant. One particular choice of the parameters that yield these conditions are γ 0 = γ 1 = 1/ N, P i = 1, i N, N r = 0, L b (i, j) = 1, (i, j) A, and L b (i, j) = 2 N, (i, j) / A. It can then be easily realized that the derived MLP is equivalent to the edge coloring problem, and the conclusion follows immediately. 6 Mathematical Formulations We study MNP and MLP using mathematical programming formulations. We first present two linear integer formulations: a node-slot formulation for MNP, and a link-slot formulation for MLP. We then formulate the two problems using set covering formulations, for which we will derive the column generation method. 6.1 A Node-Slot Formulation Let T = {1,..., T } be a set of time slots. To ensure feasibility of MNP, it is sufficient that T = N. We introduce the following binary variables. 1 if time slot t is assigned to node i, x it = 0 otherwise. 1 if time slot t is used, y t = 0 otherwise. MNP can be formulated using the following node-slot formulation (NSF). 7

8 [NSF] zn = min y t (3) t T x it 1, i N, (4) t T x it y t, i N, t T, (5) x it + x jt 1, i N, t T, (6) j:(j,i) A P i /N r L b (i, j) x it + γ 1 (1 + M ij )(1 x it ) γ 1 (1 + P k /N r L b (k, j) x kt), (i, j) A, t T, (7) k N:k i,j x it {0, 1}, i N, t T, (8) y t {0, 1}, t T. (9) The objective function (3) minimizes the total number of time slots. Constraints (4) ensure that every node is assigned at least one slot. Constraints (5) state that a slot is used (i.e., y t = 1) if it is assigned to any node. Constraints (6) ensure that different time slots are assigned to two nodes if they are the two end nodes of a link, or if both have links to a third node. The SIRcriterion is defined in (7). If slot t is not assigned to node i (i.e., x it = 0) and M ij is sufficiently large, then (7) is redundant. If x it = 1, the constraint reads P i /N r γ L b (i,j) 1(1 + k N:k i,j (2). P k /N r L b (k,j) x kt), which corresponds to the SIR-criterion To ensure that (7) is redundant when x it = 0, M ij can be set to M ij = k N:k i,j(p k /N r )/L b (k, j), i.e., the sum of the potential interference from all nodes other than i and j. However, not all nodes in the set {k N : k i, j} will transmit simultaneously, because these nodes must also satisfy constraints (6) and (7). It is therefore possible to use a smaller value of M ij, which improves the linear programming relaxation (LP-relaxation) of NSF. We refer to [1] for details of computing the smallest possible value of M ij. The formulation NSF contains two types of symmetry. Fist, there are many solutions that correspond to the same assignment, but with different time slots allocated. To break this type of symmetry, we can enforce that slot t can be used only if slot t 1 is used, by adding the constraints y t y t 1, t = 2,..., T. The second type of symmetry is related to the fact that swapping the nodes of any two slots does not affect the objective function value. Such symmetry can be partially eliminated by requiring that node i must be assigned a slot with an index less or equal to i, i.e., x it = 0, i, t : i < t. 8

9 6.2 A Link-Slot Formulation Problem MLP can be formulated using a link-slot formulation (LSF), which uses the following variables. 1 if time slot t is assigned to link (i, j), x ijt = 0 otherwise. 1 if time slot t is used, y t = 0 otherwise. 1 if node i is transmitting in time slot t, v it = 0 otherwise. Formulation LSF is stated below. [LSF] zl = min y t t T x ijt 1, (i, j) A, (10) t T x ijt y t, (i, j) A, t T, (11) x ijt + x jit 1, i N, t T, (12) j:(i,j) A j:(j,i) A x ijt v it, (i, j) A, t T, (13) P i /N r L b (i, j) x ijt + γ 1 (1 + M ij )(1 x ijt ) γ 1 (1 + P k /N r L b (k, j) v kt), (i, j) A, t T, (14) k N:k i,j x ijt {0, 1}, (i, j) A, t T, (15) v it {0, 1}, i N, t T, (16) y t {0, 1}, t T. (17) In LSF, the cardinality of T can be set to A in order to guarantee feasibility. Constraints (10) and (11) correspond to (4) and (5), respectively. Constraints (12) state that two adjacent links must be assigned different time slots. The two sets of variables, x and v, are linked using constraints (13) (note that v it has the same meaning as y it in NSF), and are used to derive the SIR-constraints (14). Similar to the case of node-oriented assignment, M ij in the SIR-constraint (14) can be set to M ij = k N:k i,j(p k /N r )/L b (k, j), 9

10 although it is possible to obtain a smaller value for this coefficient. To break the symmetry in LSF, we can add the constraints y t y t 1, t = 2,..., T, and x ijt = 0, (i, j), t : B ij < t, where B ij denotes the index of link (i, j). Formulations NSF and LSF are straightforward linear integer models. From a computational point of view, the two formulations are not suitable to use. In particular, the numbers of variables and constraints grow rapidly with respect to the network size. In our numerical experiments, a state-of-the-art solver (CPLEX 7.0, [16]) could solve NSF with up to 20 nodes to optimality, but failed to find optimal (or near-optimal) solutions for larger networks. For LSF the computational complexity is much higher. Even using an excessive amount of computing time, the solver did not manage to find a feasible integer solution for any of our test networks with more than 10 nodes. 6.3 Set Covering Formulations In this section, we reformulate MNP and MLP using set covering formulations. The LP-relaxations of the set covering formulations can be efficiently solved using a column generation method. The optimal solutions of the LPrelaxations also enable optimal or near-optimal integer solutions. An instance of a set covering problem is characterized by a finite set S, and a collection of sets C. Each member in C comprises a subset of the elements in S. A feasible solution is a set cover of S, that is, a subset C C such that every element in S belongs to at least one member of C. The set covering formulations for MNP and MLP are based on the concept of tranmission groups. Here, a transmission group is a group of nodes or a group of links that can be in simultaneous transmission, and therefore can share the same time slot. Let L N and L A be the sets of transmission groups of nodes and links, respectively. We define one binary variable for each transmission group. 1 if transmission group l is assigned a time slot, x l = 0 otherwise. The set covering formulation of MNP is stated below. [NSCF] zn = min x l (18) l L N s il x l 1, i N, (19) l L N 10

11 x l {0, 1}, l L N. (20) In NSCF, s il is an indication parameter that is one if group l contains node i, and zero otherwise. The objective function (18) minimizes the total number of assigned time slots. Constraints (19) ensure that every node belongs to at least one group that is assigned a time slot. It is apparent that MNP is a set covering problem in which S = N and C = L N. Problem MLP can be formulated using the following set covering formulation. [LSCF] z L = min x l (21) l L A s ijl x l 1, (i, j) A, (22) l L A x l {0, 1}, l L A. (23) In LSCF, parameter s ijl indicates whether link (i, j) belongs to group l (i.e., s ijl = 1 if group l contains (i, j), and zero otherwise). MLP is a set covering problem in which S = A and C = L A. The two set covering formulations have a very simple constraint structure. The complexity lies mainly in the cardinality of the two sets L N and L A. For networks of realistic size, there are huge numbers of transmission groups. However, this difficulty can be overcome using a column generation approach that exploits the structure of the two formulations. 7 A Column Generation Solution Method Originally presented in [8] and [9], column generation is a decomposition technique for solving a structured linear program (LP) with few rows but many columns (variables). Column generation decomposes the LP into a master problem and a subproblem. The master problem contains a subset of the columns. The subproblem, which is a separation problem for the dual LP, is solved to identify whether the master problem should be enlarged with additional columns or not. Column generation alternates between the master problem and the subproblem, until the former contains all the columns that are necessary for finding an optimal solution of the original LP. Column generation is especially attractive for problems that can be formulated using set covering formulations, which typically contain a huge number of columns, although very few of them are used in the optimal solution. We 11

12 note that this method has been proposed in [17] to solve the graph coloring problem, which has a similar structure to our STDMA scheduling problems. 7.1 Node-oriented Assignment To apply column generation to MNP, we consider the LP-relaxation of NSCF. z LP N = min x l (24) l L N s il x l 1, i N, (25) l L N 0 x l 1, l L N. (26) The column generation master problem is the same as the above LP-relaxation, except that the set of transmission groups, L N, is replaced by a subset L 0 N L N. To ensure feasibility of the master problem, the set L 0 N must satisfy l L 0 s il 1, i N. One particular choice of L 0 N N is the node set N (i.e., the set of transmission groups derived by TDMA). When the master problem is solved, we need to identify whether it can be improved by adding new columns (transmission groups) to L 0 N. In LP terms, this amounts to examining whether there exists any transmission group l L N, for which the corresponding variable x l has a strict negative reduced cost. Using LP-duality, the reduced cost c l of variable x l is c l = 1 i N β i s il, (27) where β i, i N, are the (optimal) dual variables to (25). Clearly, there exists at least one variable with negative reduced cost if and only if the minimum of (27) is negative. We are thus interested in the following optimization problem. min c l = 1 max l L N l L N i N β i s il. (28) The column generation subproblem is equivalent to (28), but formulated differently. We use the following variables in the subproblem. 1 if node i is included in the tranmission group, s i = 0 otherwise. The subproblem can be formulated as follows. 12

13 max i N s i + β i s i (29) j:(j,i) A s j 1, i N, (30) P i /N r L b (i, j) s i + γ 1 (1 + M ij )(1 s i ) γ 1 (1 + P k /N r L b (k, j) s k), (i, j) A, (31) k N:k i,j s i {0, 1}, i N. (32) Note the similarity between the constraints in the subproblem and those in the node-slot formulation NSF. However, the subproblem only considers one time slot for one transmission group, while in NSF transmission groups for all time slots are to be determined simultaneously. If the optimal solution to the subproblem results in a strictly negative reduced cost, the corresponding new transmission group is added to the master problem, which is then reoptimized, and the column generation method proceeds to the next iteration. Otherwise, the LP-relaxation of NSCF has been solved to optimality, and the optimum of the master problem equals zn LP. 7.2 Link-oriented Assignment The column generation method for link-oriented assignment is very similar to that for node-oriented assignment. The LP-relaxation of LSCF reads z LP L = min x l (33) l L A s ijl x l 1, (i, j) A, (34) l L A 0 x l 1, l L A. (35) The corresponding master problem is the above LP-relaxation defined for a subset of transmission groups L 0 A L A. Given the optimal solution of the master problem, finding the transmission group with minimum reduced cost is equivalent to the following optimization problem, where β ij, (i, j) A, are the (optimal) dual variables to (34). min c l = 1 max l L A l L A (i,j) A β ij s ijl. (36) 13

14 We use two sets of variables in the formulation of the subproblem. 1 if link (i, j) is included in the transmission group, s ij = 0 otherwise, 1 if node i is transmitting, v i = 0 otherwise. Using these variables, the subproblem, which is an alternative way of formulating (36), can be stated as follows. max (i,j) A j:(i,j) A s ij + β ij s ij (37) j:(j,i) A s ji 1, i N, (38) s ij v i, (i, j) A, (39) P i /N r L b (i, j) s ij + γ 1 (1 + M ij )(1 s ij ) γ 1 (1 + P k /N r L b (k, j) v k), (i, j) A, (40) k N:k i,j s ij {0, 1}, (i, j) A, (41) v i {0, 1}, i N, (42) As for the case of node-oriented assignment, the column generation method alternates between the master problem and the subproblem, until the value of (36) is non-negative. 7.3 Enhancements The performance of the column generation method depends on the computing effort of one iteration (in particular for solving the subproblem), as well as the total number of iterations before reaching optimality. Both factors become crucial if we wish to solve large-scale network instances within reasonable computing time. We propose two enhancements for accelerating the convergence of the method. Solving the subproblems, which are integer programs, may require excessive computing time, making large-scale networks out of reach of the column generation method. In fact, a straightforward implementation of the method failed 14

15 to solve the LP-relaxation of LSCF for network instances with more than 40 nodes. To overcome this difficulty, we propose the following modification to the method. Instead of solving the subproblems to optimality, a threshold value (less than zero) is used for termination control. In particular, we halt the solution process for the subproblem after a time limit. If the best solution found so far yields a reduced cost that is less or equal to the threshold value, we terminate the solution process, and add the corresponding column (which is the transmission group with the best known reduced cost so far) to the master problem. Otherwise, the threshold value is divided by a factor of two, and the solution process is resumed for another limited amount of time, after which the (new) threshold value is used for termination control. In addition, we impose a upper bound of the threshold (i.e., the threshold value is not increased if it is greater or equal to this bound). In our implementation, the time limit is set to 10 seconds, the initial threshold value for the reduced cost is 4, and the lower bound is set to 0.1. Note that the above enhancement does not compromise the solution optimality. In particular, the upper bound of the threshold value ensures that the method solves the LP-relaxations within a finite number of iterations. The second enhancement concerns the generation of maximum feasible groups, We call a transmission group maximum feasible, if the addition of any new node (or link) will make the group infeasible. Note that, for both NSCF and LSCF, there exists at least one optimal solution in which all the transmission groups are maximum feasible. By ensuring that all groups added to the master problem is maximum feasible, we attempt to minimize the number of transmission groups that the method needs to generate before reaching optimality. To find a maximal feasible group, we incorporate an additional step after solving the subproblem. Let s i, i N be a solution (not necessarily optimal) to the subproblem for node-oriented assignment. The solution can be made maximum feasible by considering the following problem, obtained by doing two modifications to the subproblem. First, we replace the objective function (29) by max i N s i, which maximizes the total number of nodes. In addition, we add the constraints s i = 1, i N : s i = 1 to the subproblem. It can be easily realized that solving this modified problem yields a maximum feasible group, for which the reduced cost is less or equal to that of the original subproblem solution. Similarly, given a subproblem solution s ij, (i, j) A to link-oriented assignment, the corresponding modified subproblem amounts to maximizing (i,j) A s ij, with the additional constraints s ij = 1, (i, j) A : s ij = 1. The above step for finding a maximum feasible group may take long computing time (in particular for link-oriented assignment). We have therefore chosen to set a time limit (10 seconds in our implementation) in this step. The best solution found within the time limit is the transmission group added to the 15

16 master problem. 8 Integer Solutions The column generation method solves the LP-relaxations of NSCF and LSCF. If some variables are fractional-valued in the LP-optimum, the solution does not represent a feasible schedule. To obtain integer solutions, enumeration schemes (such as the branch-and-price technique used in [17] for embedding column generation into a branch-and-bound framework) or heuristics are necessary. We consider two heuristic procedures for generating integer solutions. The first procedure is a straightforward post-processing step of the column generation method. Specifically, we consider the optimal integer solution for the transmission groups that have been found in the column generation process. We do this by imposing the integrality constraints to all the variables in the master problems. The resulting integer problems (which, in fact, are restricted versions of NSCF and LSCF) are then solved to optimality using a linear integer solver. We use zn IP and zl IP to denote, respectively, the numbers of time slots found for node-oriented and link-oriented assignments using this solution procedure. The second procedure for generating feasible schedules is an iterative greedy algorithm. In one iteration, the algorithm constructs a feasible transmission group, which is assigned a time slot. The algorithm for node-oriented assignment is as follows. Initially, all the nodes are stored in a list. In our implementation, the nodes are stored following the order of their indices. The first node in the list is added to the transmission group, and removed from the list. Next, the algorithms considers the second node in the list. The node is added to the group and removed from the list if all the constraints of MNP is satisfied. Continuing in this fashion, the algorithm scans through the list and adds as many nodes and possible to the group. A time slot is then assigned to the group, and the algorithm proceeds to the next iteration. We note that the time for checking the feasibility of a transmission group is polynomial in N. Consequently, the algorithm has a polynomial time complexity. Below we provide a formal description of the algorithm, where S t is the group of nodes that are assigned time slot t, and Q is the list. (1) Initialization. (a) Set S t =, t = 1,..., N. (b) Set t = 1. (c) Store the nodes in list Q. (2) Repeat until Q is empty: (a) For i = 1,..., Q : 16

17 (i) Let n i be the node at position i of Q. (ii) If S t {n i } is a feasible transmission group, set S t = S t {n i }. (b) Set Q = Q \ S t. (c) Set t = t + 1. The greedy algorithm for link-oriented assignment is identical to the above algorithm, except that all the entities are defined for links instead of nodes. We therefore leave out the explicit description of the algorithm for the case of link-oriented assignment. We use zn G and zg L to denote the numbers of time slots generated by the greedy algorithm for node-oriented and link-oriented assignments, respectively. By applying the column generation method and the two heuristic procedures to the same network instance, we are able to obtain lower and upper bounds to the optimal schedule lengths. Specifically, inequalities zn LP zn min{zn IP, zg N } and zlp L z L min{zip L, zg L } hold. 9 Numerical Results We have used six test networks of various sizes in our numerical experiments. These networks are provided by the Swedish Defense Research Agency. The numbers of the nodes and links range from 10 to 60, and from 26 to 396, respectively. For each of the test networks, the following computations have been carried out. First, we use the general linear integer solver CPLEX (version 7.0) to solve the node-slot formulation (NSF) and the link-slot formulation (LSF). Solving these two formulations led to excessive computing time for large networks, we have therefore set a time limit of 10 hours. We then apply our column generation method to solve the LP-relaxations of the set covering formulations NSCF and LSCF, and to obtain a feasible schedule using the transmission groups that have been generated. The column generation method is implemented using AMPL [6] and CPLEX. The latter is used to solve both the master problem and the subproblem. Finally, we apply the heuristic described in the previous section to compute a feasible schedule. We have conducted our experiments on a Sun UltraSparc station with a 400 MHz CPU and 1 GB RAM. 9.1 Node-oriented Assignment Computational results of node-oriented assignment are summarized in Table 1. The second column in the table shows the number of network nodes, which equals the number of time slots in the TDMA schedule. For the formulation 17

18 Table 1 Numerical results of STDMA with node-oriented assignment. N NSF (CPLEX) NSCF (Column Generation) Heuristic Slots LP Time Slots LP Iter. Time Slots (z IP N ) (zlp N ) (zg N ) N s s 10 N s s 16 N h s 21 N h s 16 N h m19s 26 N h m31s 30 NSF, the table displays the number of time slots (of the best integer solution found within the time limit), the lower bound provided by the LP-relaxation, and the computing time. Note that, if the computing time is less than the limit (10 hours), then the best integer solution has been proven to be optimal; otherwise CPLEX has either not found the optimal solution, or did not verify optimality. For the formulation NSCF, which is solved using the column generation method, we show the number of slots of the integer solution (i.e., zn IP ), the LP-bound (i.e., zn LP ), the number of column generation iterations, and the computing time. The last column in the table shows the number of time slots of the feasible schedule found by the greedy algorithm (i.e., zn G ). The results of the greedy algorithm are obtained with very little computing effort (less than a couple of seconds) for all the test networks, we have therefore not included the solution time of this algorithm in the table. Based on the results in Table 1, we make the following observations. Formulation NSF can be used to solve small network instances to optimality. However, for large networks, this formulation is clearly not computationally efficient. In particular, CPLEX did not manage to solve the problem to optimality for any of the networks with more than 20 nodes. Moreover, the LP-relaxation of NSF is very weak, when compared to the solutions found by the other methods. The set covering formulation NSCF is more computationally efficient than NSF. The LP-relaxation of this formulation can be solved efficiently using the column generation method. We observe that the LP-relaxation provides very tight lower bounds. In addition, the transmission groups generated in the column generation procedure lead to a feasible schedule that is optimal or near-optimal. For five of the six networks, the number of time slots of the feasible schedule, zn IP, is equal to the lower bound zip N, and is therefore optimal. For network N40, the two values differ by one time slot. 18

19 Table 2 Numerical results of STDMA with link-oriented assignment. A LSF (CPLEX) LSCF (Column Generation) Heuristic Slots LP Time Slots LP Iter. Time Slots (z IP L ) (zlp L ) (zg L ) N h30m s 19 N h m22s 77 N h m23s 103 N h m47s 44 N h h32m35s 108 N h h53m16s 131 Our results show that the maximum possible spatial reuse of STDMA, which can be measured as the ratio between N and zn LP, depends on the size of the network. For the networks used in our experiments, this ratio ranges from 1.0 to The greedy algorithm performs differently with respect to the network size. For networks with 30 nodes or less, the schedules found by the greedy algorithm is optimal. For the other networks the relative difference between zn G and zlp N is up to 15.4%. 9.2 Link-oriented Assignment Table 2 summarizes the computational results for link-oriented assignment. Here, the number of time slot of the TDMA schedule equals the number of links A, which is shown in the second column of the table. The other columns have the same meaning as in Table 1. In addition, we use to denote that no solution of LSF is obtained within the time limit. From an optimization point of view, scheduling of link-oriented assignment is much more challenging than that of node-oriented assignment, because the former involves a much larger solution space. We observe that, using formulation LSF, CPLEX could only solve the problem for the network with 10 nodes. For other networks, no feasible schedule could be found within the time limit. The column generation method solved the LP-relaxation of LSCF for all networks, although the solution time increased considerably when compared to the case of node-oriented assignment. The LP-bound is very close to the integer optimum. For two of the six networks, optimal schedules were found using the transmission groups generated by the column generation method. For the 19

20 other cases, the difference between z IP L and zlp L is one or two time slots. We observe that spatial reuse is achieved for all the networks. In particular, the ratio between A and zl LP varies between 1.52 and In addition, link-oriented assignment provides higher spatial reuse than node-oriented assignment. The performance of the greedy heuristic varies by network instance. For network N40, it found the best know solution (which may be optimal). For the other networks the relative difference between zl LP and zl G is up to 28.5%. 10 Conclusions and Future Work Resource optimization is a crucial issue for ad hoc networks that use STDMA for access control. A particular optimization problem concerns finding a STDMA schedule with minimum length. In this paper, we have studied this optimization problem for node-oriented and link-oriented scheduling. The optimization problem is NP -hard. However, using set covering formulations, we are able to derive a column generation method which efficiently solves the LP-relaxations. We have also evaluated two approaches for finding feasible schedules. The first approach applies integer programming to the transmission groups generated by the column generation method, and the second approach is a simple greedy algorithm. Several conclusions can be drawn from our computational study. First of all, the LP-relaxations of the set covering formulations yield very tight bounds to the minimum schedule length. These bounds are very useful for benchmarking the performance of heuristic algorithms. Secondly, using the transmission groups generated in the column generation procedure, optimal or near-optimal solutions are constantly found. Moreover, the greedy algorithm performed well for some of the cases in our experiments (particularly for node-oriented assignment). For other cases, the solutions found by the algorithm are up to 28.5% from optimality. The main advantage of this algorithm is its simplicity, which makes it an interesting candidate for distributed implementations. There are several directions for further research. One particular topic is trafficsensitive STDMA scheduling (e.g., [11], [20]), where the primary performance measure is the network throughput. We note that, using our framework of methodology, it is possible to compute the maximal network throughput of traffic-sensitive STDMA scheduling. This topic is currently under investigation. We also note that our methodology may be useful for studying the capacity regions of ad hoc networks [22]. Another very interesting topic is to develop distributed algorithms. Moreover, to be useful in practice, schedul- 20

21 ing algorithms must be able to deal with the time-varying properties of the medium, as well as the mobility of the network units. The methodology and results presented in this paper are useful for developing simple but effective heuristic algorithms for these complex problems. Acknowledgment The authors wish to thank the research group at the Department of Communication Systems, Swedish Defense Research Agency (FOI), for the technical discussions and the test data. We thank Professor Francesco Maffioli and Professor Edoardo Amaldi at Politechnico di Milano, for the discussions of the N P -hardness results. This work is partially financed by CENIIT (Center for Industrial Information Technology), Linköping Institute of Technology, Sweden. References [1] P. Björklund, P. Värbrand, D. Yuan, Resource optimization of STDMA in ad hoc networks, Technical report, Department of Science and Technology, Linköping university, Sweden, [2] I. Cidon, M. Sidi, Distributed assignment algorithms for multi-hop packet radio networks, IEEE Transactions on Computers 38 (1989) [3] I. Chlamtac, S. S. Pinter, Distributed nodes organization algorithm for channel access in a multihop dynamic radio network, IEEE Transactions on Computers 36 (1987) [4] A.-M. Chou, V. O. K. Li, Slot allocation strategies for TDMA protocols in multihop packet radio networks, IEEE INFOCOM 92, 1992, pp [5] A. Ephremides, T. Truong, Scheduling broadcasts in multihop radio networks, IEEE Transactions on Communications 38 (1990) [6] R. Fourer, D. M. Gay, W. Kernighan, AMPL - A Modeling Language for Mathematical Programming, Boyd & Fraser, Danvers, MA, [7] N. Funabiki, Y. Takefuji, A parallel algorithm for broadcast scheduling problems in packet radio networks, IEEE Transactions on Communications 41 (1993) [8] P. C. Gilmore, R. E. Gomory, A linear programming approach to the cutting stock problem, Operations Research 9 (1961) [9] P. C. Gilmore, R. E. Gomory, A linear programming approach to the cutting stock problem - part II, Operations Research 14 (1963)

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