Full exploitation of the bandwidth resources of Wireless Networks is challenging because of the sharing of the radio medium among neighboring nodes. Practical algorithms and distributed schemes that tries to optimising the use of the network radio resources. In this technical report we present the proof that maximising the network capacity is is an APX Complete problem (not approximable within 1 (1 - 2^(-k)) - eps for eps > 0).
To achieve the maximimum throughput capacity of a network several optimization problems have to be solved. @cite_8 , they investigate the problem of channel assignement in wireless network. They prove that this problem is NP-Complete and provide a polynomial-time approximation scheme (abbreviated PTAS) for this problem. The network capacity depends also on the number of time slots required to successfully schedule all links. @cite_3 , authors investigate the scheduling problem with SINR constraints, based on the physical SINR models -also called PHY graph model- @cite_18 @cite_6 , and show it to be NP-Complete. In order to prove this result, they give a polynomial time reduction from the well-known subset sum problem. The approximation corresponding problem, called Approx-Subset-Sum is an FPTAS. @cite_1 , authors also invesgate the throughput maximization problem under SINR constraints model and graph-model. They conjecture that the throughput maximization problem is NP-Complete according to the result given in @cite_3 . In this article, we prove the validity of this conjecture and moreover, we provide an additional result that is the approximation throughput maximization problem is APX-Complete. More precisly, we focus on the Remaining Capacity problem ( ) as defined in @cite_21 .
@cite_18 @cite_8 @cite_21 @cite_1 @cite_6 @cite_3
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In this work we study the problem of determining the throughput capacity of a wireless network. We propose a scheduling algorithm to achieve this capacity within an approximation factor. Our analysis is performed in the physical interference model, where nodes are arbitrarily distributed in Euclidean space. We consider the problem separately from the routing problem and the power control problem, i.e., all requests are single-hop, and all nodes transmit at a fixed power level. The existing solutions to this problem have either concentrated on special-case topologies, or presented optimality guarantees which become arbitrarily bad (linear in the number of nodes) depending on the network's topology. We propose the first scheduling algorithm with approximation guarantee independent of the topology of the network. The algorithm has a constant approximation guarantee for the problem of maximizing the number of links scheduled in one time-slot. Furthermore, we obtain a O(log n) approximation for the problem of minimizing the number of time slots needed to schedule a given set of requests. Simulation results indicate that our algorithm does not only have an exponentially better approximation ratio in theory, but also achieves superior performance in various practical network scenarios. Furthermore, we prove that the analysis of the algorithm is extendable to higher-dimensional Euclidean spaces, and to more realistic bounded-distortion spaces, induced by non-isotropic signal distortions. Finally, we show that it is NP-hard to approximate the scheduling problem to within n 1-epsiv factor, for any constant epsiv > 0, in the non-geometric SINR model, in which path-loss is independent of the Euclidean coordinates of the nodes. We give algorithms with constant-factor performance guarantees for several capacity and throughput problems in the SINR model. The algorithms are all based on a novel LP formulation for capacity problems. First, we give a new constant-factor approximation algorithm for selecting the maximum subset of links that can be scheduled simultaneously, under any non-decreasing and sublinear power assignment. For the case of uniform power, we extend this to the case of variable QoS requirements and link-dependent noise terms. Second, we approximate a problem related to cognitive radio: find a maximum set of links that can be simultaneously scheduled without affecting a given set of previously assigned links. Finally, we obtain constant-factor approximation of weighted capacity under linear power assignment. We consider the problem of resource allocation in downlink OFDMA systems for multi service and unknown environment. Due to users' mobility and intercell interference, the base station cannot predict neither the Signal to Noise Ratio (SNR) of each user in future time slots nor their probability distribution functions. In addition, the traffic is bursty in general with unknown arrival. The probability distribution functions of the SNR, channel state and traffic arrival density are then unknown. Achieving a multi service Quality of Service (QoS) while optimizing the performance of the system (e.g. total throughput) is a hard and interesting task since it depends on the unknown future traffic and SNR values. In this paper we solve this problem by modeling the multiuser queuing system as a discrete time linear dynamic system. We develop a robust H∞ controller to regulate the queues of different users. The queues and Packet Drop Rates (PDR) are controlled by proposing a minimum data rate according to the demanded service type of each user. The data rate vector proposed by the controller is then fed as a constraint to an instantaneous resource allocation framework. This instantaneous problem is formulated as a convex optimization problem for instantaneous subcarrier and power allocation decisions. Simulation results show small delays and better fairness among users. We investigate the problem of energy efficiency in TDMA link scheduling with transmission power control using a realistic SINR-based interference model, given packets of a set of links to be transmitted within a latency bound. First we formulate a fundamental optimization problem (TJSPC) that provides tunable tradeoffs between energy, throughput and latency through a single parameter β. We present both expo- nential and polynomial complexity solutions to this problem and evaluate their performance. Our results show that for moderate traffic loads, with appropriate tuning of parameters, major energy savings can be obtained without significantly sacrificing throughput. We then investigate the scheduling and power control problem with the objective of minimizing the total transmission energy cost under the constraint that all transmission requests are satisfied (JSPC-TR). We present an iterative approach to solve JSPC-TR that leverages the heuristics for TJSPC and converges rapidly to the setting of β which achieves energy efficiency while guaranteeing data delivery.