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... in those time slots to which is assigned. To guarantee the required transmission quality, a transmission over a link (i, j) can be activated only if the Signal-to-Interference and Noise Ratio (SINR) =-=[14]-=- at the receiver j is sufficiently high to correctly decode the signal, i.e. it is above a given threshold. Besides 3 i j l m psGi ij psGl lj psGm m j Figure 1: Signal-to-Interference and Noise Ratio ...

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...time. Given the exponential number of possible transmission patterns, this work presents results with small size networks or highly constrained patterns. Power Control Resource Allocation problem. In =-=[10]-=- joint scheduling and power control are considered, with the goal of maximizing network throughput. Routing is not taken into account. The general objective is to maximize network throughput. The solu...

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...chniques have been proposed to enhance the solution of optimization problem, and we refer to [19] for a recent survey. A complete introduction to CP is out of the scope of this paper, and we refer to =-=[2]-=- for an introductory text book to, and to [18] for advanced topics on the integration of CP with Integer Programming. The CP-based column generation framework used in this paper was introduced in [11]...

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...this paragraph are characterized by a significant optimality gap or by a solution quality comparison with respect to other heuristic approaches. Power and Rate Control Resource Allocation problem. In =-=[9]-=- the joint routing, scheduling, and power control problem is considered taking into account long term minimum rate requirements of traffic flows. A two steps approach is considered that optimizes sepa...

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...traffic flows. A two steps approach is considered that optimizes separately scheduling with power control and routing with rate control. The optimization objective is the total power minimization. In =-=[5]-=- this problem version is studied considering a non-linear optimization problem and the solution is found applying an approximation multi-step algorithm. In [8] power minimization routing and schedulin...

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...solutions in terms of optimality. A MILP formulation for the joint scheduling, power control, and rate control problem is proposed in [16] but the solution approach is based on a greedy algorithm. In =-=[15]-=- generic non linear formulations are proposed for routing and transmission scheduling problems. The aims is to evaluate the performance of a network in terms of throughput, rate, or fairness, rather t...

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... [2] for an introductory text book to, and to [18] for advanced topics on the integration of CP with Integer Programming. The CP-based column generation framework used in this paper was introduced in =-=[11]-=- for a crew assignment problem. The main motivation for investigating the use of Constraint Programming for solving the pricing is to exploit its strength in solving highly constrained feasibility pro...

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...nd the interference constraints which seem to be extremely challenging for MILP solvers. Along with MILP formulations, we present Constraint Programming (CP) formulations and use the CP solver Gecode =-=[12]-=- to solve the problems. The reminder of the paper is organized as follows: Section 2 introduces the problem and review the state of the art; Section 3 details the mathematical programming models; Sect...

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... to each pair of nodes that can transmit, without considering the traffic load. Some other works, even though they do not apply any column generation technique, are based on transmission patterns. In =-=[22]-=- the authors consider a reduced set of all possible fixed-power transmission patterns. This set is populated with a greedy algorithm that maximizes the number of simultaneously active links for each p...

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...f simultaneous transmissions, whereas the second one sets power to minimize the total transmitted power. The powercontrolled scheduling algorithm proposed in [3] is also based on a two steps approach; however, the set of links obtained in the first phase is usually unfeasible and it is pruned in the second phase through a greedy algorithm that directly considers SINR constraints. The authors in [4] propose a MILP model with quadratic constraints, which does not consider the routing. The optimal solution is only used to benchmark the quality of their Heuristic algorithm for small instances. In [21] a throughput maximization problem is presented within a scheduling and power control framework. The solution approach consists in solving iteratively the relaxed LP formulation of the MILP problem. At each iteration some links are fixed and the next iteration tries to add other feasible simultaneous transmissions. In [17] end-to-end routing is included in the optimization problem. The routing problem is separately solved through a shortest path approach defining a link metric based on the number of interference conflicts caused by each link. The scheduling and power control problems are then ...

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... the SINR is strictly related to the emitted power of each node, and therefore to better exploit the time resource. A version of PCRA in which the routing is not considered is proved to be NP-hard in =-=[4]-=-. Also in PCRA the transmission rate is fixed and one packet is sent for each time slot. In the third case (Power and Rate Control Resource Allocation problem – PRCRA), transmission rate can be optimi...

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...onsists in solving iteratively the relaxed LP formulation of the MILP problem. At each iteration some links are fixed and the next iteration tries to add other feasible simultaneous transmissions. In =-=[17]-=- end-to-end routing is included in the optimization problem. The routing problem is separately solved through a shortest path approach defining a link metric based on the number of interference confli...

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...lution of optimization problem, and we refer to [19] for a recent survey. A complete introduction to CP is out of the scope of this paper, and we refer to [2] for an introductory text book to, and to =-=[18]-=- for advanced topics on the integration of CP with Integer Programming. The CP-based column generation framework used in this paper was introduced in [11] for a crew assignment problem. The main motiv...

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...generation is implemented with the Gecode constraint environment [12]. Gecode is a research tool developed by an academic research group, and it used to experiment new ideas on Constraint Programming =-=[20]-=-, while CPLEX is an outstanding commercial software. Let us remark that we do tune the Cplex parameters to solve the pricing, since its default setting are far from being satisfactory. First we notice...

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...duling – i.e. assigning links to time slots – problem. In the following we present a brief survey of those closest to the problems addressed in this paper. Fixed Power Resource Allocation problem. In =-=[6]-=- several Integer Linear Programming (ILP) models for fixed-power scheduling-only optimization are presented, in particular some of these are solved by column generation methods. In the models proposed...

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...tically finds the maximum subset of simultaneous transmissions, while the second one sets power in order to minimize the total transmitted power. The power controlled scheduling algorithm proposed in =-=[3]-=- is also based on a two steps approach, however the set of links obtained in the first phase is usually unfeasible and it is pruned in the second phase through a greedy algorithm that directly conside...

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...roblems. The aims is to evaluate the performance of a network in terms of throughput, rate, or fairness, rather than providing a tool to solve real life size resource allocation problems. Finally, in =-=[7]-=- all the three versions of the problem, considering only the scheduling, are introduced. Extensive computational results obtained via a classical column generation approach are reported along with the...

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...g constraint imposing that future solutions should be better than the best one found so far. A number of techniques have been proposed to enhance the solution of optimization problem, and we refer to =-=[19]-=- for a recent survey. A complete introduction to CP is out of the scope of this paper, and we refer to [2] for an introductory text book to, and to [18] for advanced topics on the integration of CP wi...

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...es, however none of them careful assesses the quality of obtained solutions in terms of optimality. A MILP formulation for the joint scheduling, power control, and rate control problem is proposed in =-=[16]-=- but the solution approach is based on a greedy algorithm. In [15] generic non linear formulations are proposed for routing and transmission scheduling problems. The aims is to evaluate the performanc...

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...propose a MILP model with quadratic constraints, which does not consider the routing. The optimal 5 solution is only used to benchmark the quality of their heuristic algorithm for small instances. In =-=[21]-=- a throughput maximization problem is presented within a scheduling and power control framework. The solution approach consists in solving iteratively the relaxed LP formulation of the MILP problem. A...

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...tive is the total power minimization. In [5] this problem version is studied considering a non-linear optimization problem and the solution is found applying an approximation multi-step algorithm. In =-=[8]-=- power minimization routing and scheduling are decomposed in a two-phase problem. The routing is determined with an energy-aware routing protocol that discovers shortest paths with an energy metric. O...

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...s. On the one hand there are network planning issues: where to place the higher level nodes, which wireless links to set up, and in some cases which higher level nodes to connect to the wired network =-=[1]-=-. On the other hand, once the network is planned, there are the resource allocation, routing and scheduling problems, considering the fact that the network is made of wireless links and the activity o...

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... the restricted master problem, but we solve it from scratch at each iteration of the column generation. In this way, we reduce the chance of getting stuck in a degenerate solution of the master (see =-=[13]-=- for additional numerical results). Besides Cplex setting, we have to compute tight values of the big-M parameters used in linearizing the SINR constraints. In the CP-based column generation, the rest...