The algorithmic study of the scheduling problem in wireless networks in the SINR model started relatively recently, motivated by the famous paper [GK00] on network through-put of wireless networks. The state of this field evolved from experimentally validated heuristic-based algorithms to algorithms with proven worst case guarantees. Below we present the work relevant to this thesis that has been done in this process (not in the chronological order).
Complexity. In the general case, i.e. when the geometry of the network does not play a significant role in the SINR formula, the complexity of the scheduling problem is shown in [BVY04], by reducing the N P-complete Vertex Coloring problem to the scheduling problem. In [GWHW09] this result is developed further and it is shown that for the uniform power scheme, it is N P-hard to approximate the scheduling problem in the general SINR model within a factor inO(n1−), for any >0, wherenis the number of links. In the case when the geometric information is incorporated in the SINR formula (which is the case considered in this thesis) and the links are located in the Euclidean plane, N P-completeness of the scheduling problem with the uniform power scheme is proven in [GOW07], by reducing the N P-complete problemPartition to the scheduling problem. In [GW08] N P-completeness of a related problem of scheduling with analog network coding is proven. In [GOW07] it is also shown that the weighted capacity problem with the uniform power scheme isN P-complete. In [AD09] it is proven that the capacity problem with power assignment optimization is alsoN P-complete. In [HM11b]
it is shown that in general metric spaces the approximation factor of any algorithm for the capacity problem must be in Ω(2α), whereα >0 is the path loss exponent, assumed to be in (2,6) in practice. However, this result does not provide concrete hardness bounds for practical values of α. By extending the results of [GOW07], in Chapter 3 we show that it isN P-complete to approximate the scheduling problem within a factor of ρ <3/2 for a large family of power schemes, including the mean and uniform power schemes, even when all the nodes of the network are arranged on a straight line. We prove a similar complexity result for the linear power scheme in a more general setting, when the links are located in a metric space.
Approximation Algorithms. From the algorithm design perspective, there has been a considerable amount of work in the direction of heuristic-based algorithms, e.g. link removal heuristics that sequentially remove links from the given set of links while the remaining set is not feasible with any power assignment. The feasibility check of the re-maining links and calculation of the corresponding power assignment can be done based on the work [GVGZ93, BE06]. However, these algorithms have only been evaluated by means of simulations or analysis on randomly deployed sets of links, and later were shown to have poor worst-case approximation bounds [MOW07]. The first attempts to obtain algorithms with guaranteed approximation bounds for general network topologies were made in [MWZ06, GWHW09, CKM+08]. In [GOW07] and [CKM+08]O(log ∆)-approximation algorithms were presented for the scheduling and capacity problems with the uniform power scheme and the linear power scheme, respectively, where ∆ is the ratio between the longest and shortest link lengths in the given instance of the problem.
A distributed game-theoretic O(∆2α)-approximation algorithm for the capacity prob-lem with power assignment optimization was presented in [Din10]. In [GWHW09] the
authors get improved approximation algorithms for the capacity and scheduling prob-lems with the uniform power scheme: they obtain a constant factor approximation for the capacity problem which implies anO(logn)-approximation for the scheduling prob-lem. In [FKV11] the scheduling problem is considered with the linear power scheme and an algorithm is presented that schedules any given set in O(OP T + log2n) sub-sets, whereOP T is the length of the optimum solution. Using the lower bound obtained in [FKV11], in [Ton12] (Chapter4) we obtain a constant factor approximation algorithm for the scheduling problem with the linear power scheme.
In [Ton11a] (Chapter5) we use a graph-based model - the Conflict Graph model, in order to find lower bounds for the scheduling problem and provide anO(logn)-approximation algorithm for scheduling with the mean power scheme. We also show that the schedul-ing problem with the mean power scheme in the Conflict Graph model can be approxi-mated within a factor in O(1). In [KV10] a variant of the well-known distributed pro-tocol Aloha [Tan02] is presented and it is shown that the algorithm achievesO(log2 n)-approximation (with high probability) for the scheduling problem with a large class of power schemes, including the uniform, linear and mean power schemes. In [HM11a], us-ing a more delicate argument, it is proven that the approximation guarantee for the same algorithm is in fact inO(logn). In [HM11b] a greedy algorithm is presented that achieves a constant factor approximation for the capacity problem with a large family of power schemes. This result also implies an O(logn)-approximation algorithm for scheduling with these power schemes. In [Kes11,Kes12] a constant factor approximation algorithm is presented for the capacity problem with power control. This also implies anO(log n)-approximation algorithm for the scheduling problem with power control. Some papers also consider finding a solution for the capacity problem distributively [Din10,AHM12].
In [HM12] a constant factor approximation algorithm is presented for the weighted capacity problem with the linear power scheme. For other power schemes anO(logW )-approximation follows by using the constant factor )-approximation algorithms for the ordinary capacity problem [HM11b], where W is the maximum weight. Our results in [Ton13b] (see Chapter6) imply anO(logn)-approximation algorithm for the weighted capacity problem with the mean power scheme.
Comparing Power Schemes. It has been shown [MW06, FKRV09] that for the scheduling problem with power control, when using power schemes depending only on the length of the links, e.g. the linear, uniform or mean power schemes, then the worst case approximation ratio depends on the topology of the network and in some cases can be very large. However, the worst case examples presented are largely unrealistic. It can be shown that when using the linear or uniform power schemes the approximation factor for this problem is inO(log ∆) [FKV11,Hal09,Ton12,AD09,CKM+08] and this
is tight in terms of ∆. When using the mean power scheme, the approximation factor is in O(log log ∆ logn) [Hal12] (see also Chapter 5 for another proof). In [HHMW13]
it is shown that when the mean power scheme is used in the capacity problem with power control the approximation factor is in O(log log ∆) (we obtained the same result in Chapter 7 using a different method based on Conflict Graphs). From the bounds presented in [FKV11, Hal09, Ton12, AD09, CKM+08, Hal12, HHMW13] it should be evident that in terms of the scheduling and capacity problems the mean power scheme should perform better than the linear or uniform power schemes. However, it is not clear if the mean power scheme performs better than the two other power schemes onevery problem instance. In [Ton11b] we show that for the capacity problem the uniform and linear power schemes yield optimal solutions of the same order for any given problem instance, and there is a large family of power schemes, including the mean power scheme, that for any problem instance achieves asymptotically not worse (and for some instances much better) solutions than the uniform or linear power schemes. Using the Conflict Graph framework, we also show that under certain restrictions on the path loss exponent the mean power scheme obtains better solutions for the scheduling problem too. This is also demonstrated by simulations [BKKV12].
Conflict Graphs. Over the course of the research on the scheduling problem different models for conflict graphs have been considered for modeling the communication, mostly based ondisk-graphs. Depending on the way a conflict graph is constructed, the solutions obtained for the scheduling problem in this model can have two kinds of drawbacks.
First, because of an oversimplified modeling of signal propagation, conflict graphs based on disk-graphs do not use the space economically, thus obtaining longer solutions than one could obtain in the SINR model. This was demonstrated both analytically and by means of experiments in [MWY06, CKM+08, GH01]. However, this issue can be eliminated by properly designing the conflict graph. A more significant drawback of graph-based models stems from the fact that these models do not take into account the cumulative nature of interference, i.e. in this case the solutions obtained in a graph based model can be overoptimistic and can be significantlyshorter than solutions obtained in the SINR model. This issue was discussed in [BR03] and was demonstrated through simulations. Another paper considering this problem is [KIR09].
Even though graph-based models have so many drawbacks, their simple structure and rich algorithmic possibilities of graph theory make it tempting to consider them. A kind of a conflict graph, which is based on the SINR formula, is introduced in [Hal09,Hal12], and it is shown that when all the links in consideration have similar lengths the conflict graph model and the SINR model yield similar solutions for the scheduling and capacity problems. In [Ton13a] (see Chapter6) we consider another kind of conflict graphs, which in this thesis is simply called Conflict Graph model, which is also based on the SINR
formula. The vertices of the graph are the links and two links are connected by an edge, i.e. they are conflicting, if at least one of them would make “too much” interference for the other one if they transmitted in the same time slot. A different Conflict Graph is defined for each power scheme, which is a supergraph of the corresponding graph defined in [Hal09, Hal12]. The length of an optimal solution of the scheduling problem in the conflict graph is a lower bound on the length of an optimal solution in the SINR model. It is shown that the difference between the lengths of solutions in the Conflict Graph model and the SINR model can be up to Θ(log ∆) (thus, up to Θ(n) in terms of the number of links) when the uniform or linear power scheme is used and the links are allowed to have arbitrary lengths. However, it is shown that when the mean power scheme is used in the same setting this difference is just in O(logn). Moreover, in [Ton13b] we show that when the capacity problem is considered the difference between the two models is inO(1). This gives the possibility to treat the SINR model more easily, using the graph framework. Some of the main results in this thesis are obtained using this relationship between the two models.
The Bidirectional SINR Model. The bidirectional SINR model, which was intro-duced in [FKRV09], is a variant of the SINR model where it is required that if a link is assigned to a time slot it must be feasible on both directions of communication. Many algorithms and relations obtained for the SINR model can be reformulated to hold for the bidirectional model [Hal09,Ton11a,Ton11b,Kes11]. Since the bidirectional model is more constrained, it yields shorter solutions for the scheduling problem. However, in [Ton11b] we show that the difference between the two models in the case of the uni-form and linear power schemes is inO(1), and in the case of the mean power scheme is in O(logn) when the scheduling problem is considered, and is inO(1) for the three power schemes when the capacity problem is considered. We also show that the difference be-tween the optimal solutions with power optimization in the two models is inO(log log ∆) in the case of the capacity problem (which is a tight bound) and inO(log log ∆ logn) in the case of the scheduling problem.