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).
@cite_18: When n identical randomly located nodes, each capable of transmitting at W bits per second and using a fixed range, form a wireless network, the throughput spl lambda (n) obtainable by each node for a randomly chosen destination is spl Theta (W spl radic (nlogn)) bits per second under a noninterference protocol. If the nodes are optimally placed in a disk of unit area, traffic patterns are optimally assigned, and each transmission's range is optimally chosen, the bit-distance product that can be transported by the network per second is spl Theta (W spl radic An) bit-meters per second. Thus even under optimal circumstances, the throughput is only spl Theta (W spl radic n) bits per second for each node for a destination nonvanishingly far away. Similar results also hold under an alternate physical model where a required signal-to-interference ratio is specified for successful receptions. Fundamentally, it is the need for every node all over the domain to share whatever portion of the channel it is utilizing with nodes in its local neighborhood that is the reason for the constriction in capacity. Splitting the channel into several subchannels does not change any of the results. Some implications may be worth considering by designers. Since the throughput furnished to each user diminishes to zero as the number of users is increased, perhaps networks connecting smaller numbers of users, or featuring connections mostly with nearby neighbors, may be more likely to be find acceptance.
@cite_8: Multichannel communication in a wireless mesh network with routers having multiple radio interfaces significantly enhances the network capacity. Efficient channel assignment and routing is critical for realization of optimal throughput in such networks. In this paper, we investigate the problem of finding the largest number of links that can be activated simultaneously in a wireless mesh network subject to interference, radio and connectivity constraints. Our goal is to activate all such links and we present an interference aware channel assignment algorithm that realizes this goal. We show that the Link Interference Graph created by utilizing a frequently used interference model gives rise to a special class of graphs, known as overlapping double-disk (ODD) graphs. We prove that the Maximum Independent Set computation problem is NP-complete for this special class of graphs. We provide a Polynomial Time Approximation Scheme (PTAS) for computation of the Maximum Independent Set of an ODD graph. We use this PTAS to develop a channel assignment algorithm for a multiradio multichannel Wireless Mesh Network. We evaluate the performance of our channel assignment algorithm by comparing it with the optimal solution obtained by solving an integer linear program. Experimental results demonstrate that our channel assignment algorithm produces near optimal solution in almost all instances of the problem.
@cite_21: We show that link interferences in multihop wireless networks make the problem of selecting a path satisfying bandwidth requirements an NP-complete problem, even under simplified rules for bandwidth reservation. This is in sharp contrast to path selection in wireline networks where efficient polynomial algorithms exist. We also describe a distributed mechanism for the problem of slot allocation according to bandwidth reservation in a wireless slotted environment.
@cite_1: A fundamental problem in wireless networks is to estimate its throughput capacity - given a set of wireless nodes, and a set of connections, what is the maximum rate at which data can be sent on these connections. Most of the research in this direction has focused on either random distributions of points, or has assumed simple graph-based models for wireless interference. In this paper, we study capacity estimation problem using the more general Signal to Interference Plus Noise Ratio (SINR) model for interference, on arbitrary wireless networks. The problem becomes much harder in this setting, because of the non-locality of the SINR model. Recent work by (2006) has shown that the throughput in this model can differ from graph based models significantly. We develop polynomial time algorithms to provably approximate the total throughput in this setting.
@cite_6: To date, topology control in wireless ad hoc and sensor networks--the study of how to compute from the given communication network a subgraph with certain beneficial properties .has been considered as a static problem only; the time required to actually schedule the links of a computed topology without message collision was generally ignored. In this paper we analyze topology control in the context of the physical Signal-to-Interference-plus-Noise-Ratio (SINR) model, focusing on the question of how and how fast the links of a resulting topology can actually be realized over time.For this purpose, we define and study a generalized version of the SINR model and obtain theoretical upper bounds on the scheduling complexity of arbitrary topologies in wireless networks. Specifically, we prove that even in worst-case networks, if the signals are transmitted with correctly assigned transmission power levels, the number of time slots required to successfully schedule all links of an arbitrary topology is proportional to the squared logarithm of the number of network nodes times a previously defined static interference measure Interestingly, although originally considered without explicit accounting for signal collision in the SINR model, this static interference measure plays an important role in the analysis of link scheduling with physical link interference. Our result thus bridges the gap between static graph-based interference models and the physical SINR model. Based on these results, we also show that when it comes to scheduling, requiring the communication links to be symmetric may imply significantly higher costs as opposed to topologies allowing unidirectional links.
@cite_3: In this paper we study the problem of scheduling wireless links in the geometric SINR model, which explicitly uses the fact that nodes are distributed in the Euclidean plane. We present the first NP-completeness proofs in such a model. In particular, we prove two problems to be NP-complete: Scheduling and One-Shot Scheduling. The first problem consists in finding a minimum-length schedule for a given set of links. The second problem receives a weighted set of links as input and consists in finding a maximum-weight subset of links to be scheduled simultaneously in one shot. In addition to the complexity proofs, we devise an approximation algorithm for each problem.
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 .