Bounds and Algorithms for Scheduling Transmissions in Wireless Networks

Thesis

Reference

Bounds and Algorithms for Scheduling Transmissions in Wireless Networks

TONOYAN, Tigran

Abstract

The topic of the thesis is medium access which is one of the fundamental problems in wireless networks: as a result of the spatial nature of wireless signals wireless transmissions experience interference from each other, thus making spatially close simultaneous transmissions conflict with each other. A naturally arising problem in this context is the problem of scheduling where the transmission requests must be organized into a small number of subgroups each of which can transmit simultaneously in one time slot, the ultimate goal being executing all transmission in the minimum time. The thesis considers scheduling

problems and related problems in the well known SINR

(signal-to-interference-and-noise-ratio) model and extends the current state of the art on the treatment of these problems from a theoretical point of view. Several complexity results and approximation algorithms are presented for the aforementioned problems. A graph-based framework is introduced which is used to obtain results for medium access problems in the SINR model, as well as to extend the idealistic geometric SINR model towards more realistic [...]

TONOYAN, Tigran. Bounds and Algorithms for Scheduling Transmissions in Wireless Networks. Thèse de doctorat : Univ. Genève, 2013, no. Sc. 4616

URN : urn:nbn:ch:unige-322402

DOI : 10.13097/archive-ouverte/unige:32240

Available at:

http://archive-ouverte.unige.ch/unige:32240

Disclaimer: layout of this document may differ from the published version.

1 / 1

D´epartement d’informatique Professeur Jos´e Rolim

Bounds and Algorithms for Scheduling Transmissions in Wireless Networks

TH` ESE

pr´esent´ee `a la Facult´e des sciences de l’Universit´e de Gen`eve pour obtenir le grade de Docteur `es sciences, mention informatique

par

Tigran Tonoyan

de

Erevan, Arm´enie

Th`ese N^o 4616

GEN`EVE

Atelier d’Impression Uni-Mail 2013

you have long taken for granted.”

Bertrand Russell

Avec la croissance rapide des technologies wireless, une des questions fondamentales des r´eseaux sans fils, les acc`es aux supports, sont pass´es au premier plan. Les appareils sans fils travaillant dans la mˆeme bande de fr´equences doivent coordonner leurs acc`es au spectre, leurs niveaux d’´energie et ´eviter les collisions lors des transmissions `a cause des interf´erences, afin de maintenir un fonctionnement du r´eseau de bonne qualit´e. La ques- tion r´eside sp´ecialement dans les r´eseau sans fils `a cause de la nature spatiale des signaux wireless, contrairement aux transmissions cˆabl´ees dans les r´eseaux cabl´es. Le traitement th´eorique des probl`emes li´es `a l’acc`es aux supports d´epend de mani`ere cruciale du mod`ele de communication adopt´e. Ces dix derni`eres ann´ees, le mod`ele SINR (Signal to Inter- ference and Noise Ratio) est devenu populaire dans la communaut´e algorithmique et a remplac´e les mod`eles plus simplistes bas´es sur les graphes qui ne prennent pas en compte la nature additive des interf´erences. Cependant, dans sa forme la plus g´en´erale, le mod`ele SINR ne peut pas ˆetre mani´ee algorithmiquement. C’est la raison pour laque- lle on consid`ere principalement des variantes g´eom´etriques du mod`ele. Par contre, cette simplification a un prix: on doit supposer un environnement ”id´eal”, des antennes om- nidirectionnelles et aucun obstacle entre les supports de transmission. Mˆeme dans ces mod`eles simplifi´es, tous les probl`emes int´eressants restent durs `a r´esoudre exactement, et, une des options pour les traiter est de rechercher des algorithmes d’approximation. Dans cette th`ese, nous consid´erons deux familles de probl`emes d’optimisation, le probl`eme d’ordonnancement et le probl`eme de capacit´e qui d´ecrit l’acc`es au support. Ces dix derni`eres ann´ees, ces probl`emes ont ´et´e ´etudi´es avec le mod`ele g´eom´etrique SINR, et les techniques d´evelopp´ees ont ´et´e g´en´eralis´ees pour la variante m´etrique du mod`ele.

Ce dernier cependant coupe tous les liens avec les mod`eles bas´es sur les graphes. Nous essayons au contraire d’´etablir des connections entre le mod`ele SINR et certains mod`eles bas´es sur les graphes. Il s’av`ere que lorsque les niveaux d’´energie des appareils de trans- mission sont ajust´es correctement, Les performances d’un syst`eme dans un mod`ele SINR peuvent ˆetre bien approxim´ees par un mod`ele bas´es sur les graphes. Ceci nous ouvre la porte de la th´eorie des graphes pour les probl`emes bas´es SINR et nous aide a utiliser des m´ethodes bas´ees sur les graphes afin d’obtenir une bonne approximation pour les mod`eles SINR. Grace `a ces techniques, nous sommes aussi capables de comparer les per- formances de diff´erentes strat´egies d’affectation de puissance en termes de planification, probl`emes de capacit´e et estimation de la meilleure performance possible du r´eseau. La connection entre ces deux mod`eles peut mˆeme aider `a ´etendre le mod`ele g´eom´etrique SINR ”id´eal”, en permettant de mod´eliser des param`etres plus complexes comme des obstacles dans le r´eseau et obtenir des solutions non triviales.

With the rapid growth of wireless technology, one of the fundamental issues in wireless networks - medium access, has moved into the foreground. Wireless devices working in the same frequency range need to coordinate their access to the spectrum and their power levels and avoid transmission collisions because of interference, in order to main- tain a high quality operation of the network. This issue stands especially sharply in wireless networks because of the spatial nature of wireless signals, as opposed to wired transmissions in wired networks. Theoretical treatment of problems related to medium access depends crucially on the adopted model of communication. In the last decade the SINR (Signal to Interference and Noise Ratio) model has become popular in the algo- rithmic community, and has come to replace more simplistic graph-based models, which do not take into account the additive nature of signal interference. However, in its most general form the SINR model cannot be handled algorithmically. This is the reason why mainly geometric variants of the model are considered. But this simplification comes at a price: one has to assume an “ideal” environment, omni-directional antennas and no obstacles between the transmitting devices. Even in this simplified model, all inter- esting problems remain hard to solve exactly, and one of the options for dealing with these problems is to search for approximation algorithms. In this thesis we consider two families of optimization problems, the scheduling problem and the capacity problem, which describe medium access. In the last decade these problems have been studied in the geometrical SINR model, and the developed techniques have been generalized to the metric variant of the model. The latter, however, cuts all the ties with graph-based models. We, instead, try to establish connections between the SINR model and certain graph-based models. It turns out that when the power levels of transmitting devices are adjusted properly, the performance of a system in the SINR model can be well ap- proximated by a graph-based model. This opens the doors of graph theory in front of SINR-based problems and helps us to use graph-based methods in order to obtain good approximations for problems in the SINR model. Using these techniques, we are also able to compare the performance of different power assignment strategies in terms of the scheduling and capacity problems and estimate the best possible performance of the network. The connection between the two models even helps to extend the “ideal”

geometric SINR model, by allowing to model more complex settings such as obstacles in the network, and still get non-trivial solutions.

I would like to express my gratitude to my advisor Jos´e Rolim for giving me the oppor- tunity to work on this project, and for his support. I thank all my colleagues from TCS Sensor Lab of the University of Geneva for the nice atmosphere that I have had here during my years as a PhD student. Special thanks to Aubin Jarry, Pierre Leone and Christoforos Raptopoulos for some insightful discussions on the topics of my thesis, to Eugenio Noto for helping with the organization of the thesis, and to my close friend and colleague Hakob Aslanyan who has helped me a lot during these years. I will always be grateful to my lovely fianc´ee Ilona for her patience and constant support.

iv

Contents

R´esum´e ii

Abstract iii

Acknowledgements iv

List of Figures viii

Symbols ix

1 Introduction 1

1.1 Related Work . . . 3

1.2 Thesis Overview . . . 7

2 Definitions and Preliminaries 11 2.1 The Complexity of Problems and Approximation Algorithms . . . 11

2.2 Graphs. . . 12

2.2.1 Inductive Independence Number . . . 13

2.3 Metric Spaces . . . 15

2.3.1 Doubling Metric Spaces . . . 16

2.3.2 Euclidean Spaces and the Lebesgue Measure . . . 17

2.4 Capacity and Scheduling Problems . . . 17

2.4.1 The Path-Loss Model for Signal Decay and Cumulative Interference 19 2.4.2 SINR Model of Signal Reception . . . 20

2.4.3 ISR and Zero Noise . . . 21

2.5 The Conflict Graph Model . . . 25

2.6 Power Schemes . . . 27

2.6.1 Ordering Power Schemes. . . 28

3 The Complexity of Scheduling 30 3.1 Complexity of Scheduling on the Line with Sub-Linear Power Schemes . . 30

3.2 Complexity of Scheduling in Metric Spaces with the Linear Power Scheme 34 4 The ISR-Based Lower Bound for the Scheduling Problem 37 4.1 ISR-based Lower Bounds . . . 38

v

4.2 A Greedy Algorithm Scheme . . . 40

4.2.1 Analysis of Algorithm 4.1 for the Linear Power Scheme . . . 41

4.2.1.1 Feasibility of the Output . . . 41

4.2.1.2 The Approximation Factor . . . 45

4.2.2 Feasibility of Algorithm 4.1 for Super-Linear Power Schemes . . . 46

4.3 Lower Bounds for Non-Linear Power Schemes . . . 47

4.3.1 Increasing Sub-Linear Power Schemes . . . 47

4.3.2 Decreasing Sub-Linear Power Schemes . . . 48

4.3.3 Super-Linear Power Schemes . . . 48

5 The Graph-Based Lower Bound for the Scheduling Problem 51 5.1 Scheduling in the Conflict Graph Model . . . 51

5.1.1 A Scaling Theorem forG^γ_M(Γ) . . . 56

5.1.2 Comparing the Mean Power Scheme with Optimal Power Assign- ments . . . 57

5.2 SINR Scheduling with the Mean Power Scheme . . . 58

5.2.1 Scheduling an Independent Set . . . 59

5.2.2 Comparing the Mean Power Scheme with Optimal Power Assign- ments . . . 61

6 The Relationship Between the SINR and Conflict Graph Models 63 6.1 Sets of Similar Length Links. . . 64

6.2 The Uniform and Linear Power Schemes . . . 65

6.3 The Mean Power Scheme . . . 67

6.3.1 An Application: The Weighted Capacity of the Mean Power Scheme 72 6.4 The Gap between the Two Models in General Metric Spaces. . . 74

7 Comparing Different Power Schemes in the SINR Model 77 7.1 Comparing Capacities . . . 78

7.1.1 Conjugate Power Schemes . . . 78

7.1.2 The Capacity of the Uniform Power Scheme . . . 82

7.1.3 Comparing the Capacity of the Mean Power Scheme with the Best Achievable Capacity . . . 84

7.2 Comparing SINR Schedules Using Conflict Graphs . . . 85

8 A Modification and an Extension of the SINR Model 89 8.1 The Bidirectional SINR Model . . . 90

8.1.1 Comparing the Directed and Bidirectional SINR Models . . . 91

8.1.1.1 Given Power Schemes . . . 91

8.1.1.2 Best Achievable Capacities . . . 93

8.1.2 Applying the Results Obtained for the Directed Model to the Bidi- rectional Model . . . 94

8.2 Extending the Geometric SINR Model . . . 96

8.2.1 Extended SINR for the Mean Power Scheme . . . 96

8.2.2 Extended SINR for the Optimal Power Assignments . . . 97

9 Conclusions 100

A Lemmas from Calculus and Inequalities 102 A.1 Bounding a Series by an Integral . . . 102 A.2 An Inequality forf(x) =x^t . . . 103 A.3 Number Inequalities . . . 103

Bibliography 104

List of Figures

2.1 Doubling property ofR². . . 17

2.2 A set of links. . . 18

2.3 Path loss and interference. . . 20

2.4 Disk Graph model. . . 25

3.1 The network example for Theorem 3.1. . . 32

4.1 An example configuration for Lemma 4.5, showing B(s_i, δl_i), R₁ and nodes from SR1 with balls B(s_j, ρ_j) for each of them. The two dashed circles show the extended ringR⁰₁. . . 43

4.2 The network example for Lemma 4.14. . . 49

5.1 Illustration of (5.1). . . 53

5.2 Illustration for Lemma 5.3. By (c), we have only two types of links, belonging to setT1 orT2: we choose the larger of those sets so as to have at least a half of the set T. . . 54

6.1 The construction in Theorem 6.2. . . 66

6.2 An illustration of (6.7). . . 70

6.3 A γ-independent set of similar length links in a metric space. . . 75

A.1 Bounding a series by an integral. . . 102

viii

Symbols

m the dimension of the metric space

d(a, b) the metric distance between points aand b R^m the m-dimensional Euclidean space

δG(v) the degree of vertex v in graphG τ(G) the independence number of graph G χ(G) the chromatic index of graph G

P the class of deterministically polynomially solvable decision problems N P the class of non-deterministically polynomially solvable decision problems Γ the input set of links for scheduling and capacity problems

n the number of links in Γ P(i) the power level of link i

N_i the ambient noise around the receiver of link i β the SINR threshold value

l_i the length of link i

d_ij the distance between the sender of link iand receiver of link j d_ij the minimum distance between the nodes of link iand linkj OP T C_P(Γ) the optimum capacity of a set Γ w.r.t. power assignment P OP T S_P(Γ) the optimum schedule length of a set Γ w.r.t. power assignment P OP T S(Γ) the supremum of OP T S_P(Γ) over all power assignments P for the set Γ OP T C(Γ) the supremum of OP T C_P(Γ) over all power assignmentsP for the set Γ OP T W_P(Γ) the optimum weighted capacity of a set Γ w.r.t. power assignment P OP T W(Γ) the supremum of OP T W_P(Γ) over all power assignments P for the set Γ ISR_P(S, i) the interference to signal ratio of linkiby links S

OP T C_P(Γ) same as OP T C_P(Γ) but in the bidirectional SINR model OP T S_P(Γ) same as OP T S_P(Γ) but in the bidirectional SINR model

ix

ISR_P(S, i) same as ISR_P(S, i) but in the bidirectional SINR model

G^γ_P(Γ) the γ-conflict graph for a set of links Γ w.r.t. power assignmentP L_t the class of power schemes of the form L_t(i) =cl^tα_i

U the uniform power scheme L the linear power scheme

M the mean power scheme

S[i⁺] the subset of links in S that are longer than the linki S[i⁻] the subset of links in S that are shorter than the linki

∆ the ratio of the lengths of the longest and the shortest links

xi

Chapter 1

Introduction

The need to be able to access information at any time and anywhere together with the development of technology has made wireless communication extremely widespread and has resulted in the tremendous growth of cellular telephony. With this popularization of wireless technology, wireless services other than cellular telephony such as Wireless Local Area Networks (WLANs), ad hoc networks, sensor networks, emerged. Such a heavy use of wireless medium brings to the front the fundamental issues of wireless networks. One of such issues ismedium access to wireless medium. The increasing number of wireless services makes wireless medium a scarce resource that should be managed carefully in order to make the system meet the quality requirements. Wireless devices working in the same frequency range have to transmit in a coordinated manner so as to avoid transmission failures because of highinterference among different transmitting devices.

The algorithmic aspect of the medium access is rather multifaceted, due to numerous applications and scenarios of wireless networks. A challenging task is modeling the communication and interference. One needs reasonably treatable simple models in order to obtain non-trivial theoretical guarantees on the network behavior, but on the other hand, these models must be as close to the practice as possible.

Graph-based models are one way of modeling interference and communication in a wire- less network [KMR01,KMPS05]. In these models one definesconflicting pairs transmis- sion requests, and two transmissions can be done in the same time slot if and only if they are not conflicting. This allows to model medium access problems using conflict graphs constructed over sets of transmission requests. These models are reasonably simple and flexible, and can be treated using the reach framework of graph theory, but they are overoptimistic in modeling interference; graph-based models do not take into account theaccumulative nature of interference, which might lead to unrealistic theoretical pre- dictions [MWY06].

1

Another way of modeling wireless communication is the SINR (Signal to Noise and Interference Ratio)model. In this model, a link can transmit successfully in the presence of transmission from a set of other links if and only if

signal

interf erence+noise ≥threshold,

where signal is the power of the signal of the given link, interf erence is the total cu- mulative interference from the other transmitting links and noiseis the environmental noise. The SINR model can take into account real world phenomena like path loss and accumulative interference. However, this model is more complex and is algorithmically intractable in its most general form. This is the reason that most of the algorithmic research deals with simplifications of the SINR model, by assuming a homogeneous envi- ronment for the signal propagation, omni-directional antennas and absence of obstacles.

This thesis investigates the properties of this model and its relationship with an under- lying conflict graph model, and tries to find fundamental laws concerning the medium access problem in these simple models, as well as to extend them.

We consider the following basic optimization problems which describe medium access.

In the capacity problem, given is a set of one hop communication requests - links, each using a certain power level for transmission, and the goal is to select a maximum cardinality subset of links that is feasible with respect to the given power assignment, i.e. all the transmissions in this subset ought to be done successfully in the same time slot. Another related problem is the scheduling problem, where the goal is to partition the given set of links into the minimum number of feasible subsets, i.e. find the shortest schedule for the given set of links. These problems are derived from the consideration of the fundamental measure of system throughput of wireless networks. In contrast to the averaged measurement of the throughput, the capacity and scheduling problems aim to find specific optimal configurations of links for each particular geometrical placement of links and for the given power assignment. The capacity and scheduling problems can also be formulated in the power-control scenario, where one is also free to optimize the power assignment of the links in order to get better solutions. The weighted capacity problem, where the links are assigned weights and the sum of weights of a feasible set is optimized, can also have a practical significance, e.g. in the framework of spectrum auctions.

The main goal of this thesis is to investigate the properties of different power assign- ment policies orpower schemes with respect to the capacity and scheduling problems in the SINR model. The power schemes are simple, typically localized and generic power assignment rules. A particular power scheme can be chosen by different considerations such as energy efficiency, power control capabilities of the transmitters etc. Even though

we obtain some results for large classes of power schemes, we are especially interested in three power schemes in this thesis: theuniformpower scheme, the linear power scheme and the mean power scheme. The uniform power scheme assigns the same power level to all the links. This power scheme is used in scenarios where power control is not possible or is not desirable. The linear power scheme assigns the power levels in a way that the signal levels at the receivers of all links are equal. This is the minimal power assignment to win the environmental noise by a given factor; hence, it may be used in applications where the wireless nodes are energy constrained. The mean power scheme has been considered relatively recently due to its efficiency in terms of the scheduling problem. Nevertheless, in this thesis we are concerned only by the efficiency of power schemes with regard to the capacity and scheduling problems. One task that we consider is to design efficient algorithms for solving these problems with respect to a given power scheme. However, even in the simplified SINR model considered, these optimization problems are still computationally hard to solve exactly (in terms of the standard com- plexity theory) for practically all interesting power schemes. This leads to the search of approximation algorithms for these problems. Another aspect discussed in this thesis is comparing different power schemes in terms of optimal solutions of the problems con- sidered. We show that certain power schemes can yield better solutions for the capacity and scheduling problems than others forany given instance of these problems.

We consider the capacity and scheduling problems both in the SINR model and a graph- based model that can be derived from the SINR model, although our main interest is in the former. We consider the graph-based model mainly in the context of its rela- tionship with the SINR model. Also, combination of the two models gives possibility to incorporate obstacles in the model, which is a crucial issue for the “ideal” SINR model.

The majority of our results are based on geometric arguments involving the properties of the metric space where the wireless nodes are placed; however, several of the key results are obtained using the properties of graphs corresponding to the graph-based model mentioned above.

1.1 Related Work

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(2^α), 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(log²n)- 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(logn)- 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.

1.2 Thesis Overview

The main results of the thesis are presented in Chapters 3 to 8. In Chapter 2, the main definitions and preliminary results are presented, that are used throughout the thesis. In particular, the frameworks of metric spaces, complexity theory, graph theory are briefly introduced, as well as the main definitions of the SINR model and the Conflict Graph model.

In Chapter 3, we extend the work of [GOW07] to show that for many interesting power schemes the scheduling problem isN P-hard and cannot be approximated within a factor of ρ < 3/2 under standard assumptions of complexity theory, even when all the nodes of the network are arranged on a straight line. Thus, most probably, the best we can do is to find constant factor approximation algorithms for the scheduling problem with factors above 3/2.

In Chapter 4, we consider a lower-bound for the scheduling problem which was in- troduced in [FKV11, KV10]. We present a greedy procedure that produces potential solutions that match this lower bound for a large class of power schemes; however, the output of the algorithm is not guaranteed to be feasible. We show that in the case of the linear power scheme the algorithm always produces feasible solutions and thus it is a constant factor approximation algorithm for the linear power scheme. This is the only known constant factor approximation algorithm for the scheduling problem in the SINR model with sets of links arbitrarily placed in Euclidean spaces. For other power schemes, we show that the lower bound can be arbitrarily far from the optimum.

In Chapter5, we consider another lower-bounding technique for the scheduling problem, based on Conflict Graphs. The chromatic index of a Conflict Graph corresponds to the optimal solution of the scheduling problem in the Conflict Graph model and is naturally a lower bound for scheduling in the SINR model. The first part of Chapter5demonstrates some properties of Conflict Graphs corresponding to the mean power scheme. We show, among other things, that there is a constant factor approximation algorithm for the capacity and scheduling problem in these graph models. In the second part we apply the results obtained for Conflict Graphs in order to get an O(logn) approximation algorithm for the scheduling problem with the mean power scheme in the SINR model.

Chapter 6is dedicated to a deeper exploration of the relationship between the Conflict Graph model and the SINR model. We try to answer the question how far are solutions of the capacity and scheduling problems obtained for the same set of links in these two models. This question is considered for the uniform, linear and mean power schemes. A close connection between the two models would give the possibility to apply the tech- niques developed for the graph-based models to the scheduling and capacity problems in the SINR model. It is known that the two models yield similar solutions when the links in the given instances of the scheduling and capacity problems are of similar lengths.

We show that when arbitrary length links are allowed and either the linear of uniform power scheme is used the difference between the solutions obtained in the two models can be largely different in the worst case (in terms of ∆ it can be Θ(log ∆) and in terms of n it can be Θ(n)). In the case of the mean power scheme, however, there is a close

relationship between the two models. We show that the solutions of the capacity prob- lem in the two models differ by only an O(1) factor when the mean power scheme is used. This also implies an O(logn)-factor difference for the scheduling problem in the worst case. This relation shows that the mean power scheme is “scalable” between the two models, and allows to use the tools of graph theory for approximating problems in the SINR model.

In Chapter 7 we compare the performance of different power schemes in terms of the scheduling and capacity problems. We try to find pairs of power schemes where one of them consistently performs better than the other one. It turns out that there exist such pairs of power schemes. We show that the linear and uniform power schemes yield similar (differing by factors in O(1)) solutions for the capacity problem for any given instance of the problem. Using this result, we are able to show that there is a large class of power schemes, including the mean power scheme, which never (meaning for no instance of the problem) perform worse for the capacity problem than the uniform and linear power schemes. On some network instances these power schemes perform strictly better than the uniform and linear power schemes. Note that comparison of a pair of power schemes in terms of the capacity problem does not imply a similar relationship in terms of the scheduling problem (the opposite way implication holds). However, using the framework developed for the Conflict Graph model, we are able to show that under certain assumptions on the metric space, the mean power scheme never performs worse than both the uniform and linear power schemes.

Chapter 8 consists of two parts. In the first part we consider the bidirectional SINR model introduced in [FKRV09]. This model is more constrained and, in general, less links can transmit in the same time slot in this model, compared to the ordinary SINR model. However, we note that most of the results obtained in the other chapters hold for the bidirectional SINR model as well. We also show that for any given instance of the capacity problem the solution obtained in the bidirectional model with the best power assignment is at most O(log log ∆) factor worse than the best solution obtained in the ordinary SINR model. We show also that this bound is tight.

In the second part of Chapter8we introduce an extension of the geometric SINR model, which uses the results on the relationship between the SINR and Conflict Graph models (Chapter6), in order to model such settings that are not captured by either of the two models, but fall “in between” the two models. Such a generalization can be interpreted as inserting obstacles between the links. We show that in this general model the mean power scheme yields solutions for the capacity problem that are of the same order as in the case of the Conflict Graph model or the SINR model. We also show that these

solutions obtained by using the mean power scheme are very close to the best possible solutions with any power assignment in this extended model.

The conclusions and our perspective of the future work is presented in Chapter 9.

The main results of the thesis are covered by or follow from publications [Ton11a, Ton11b,Ton11c,Ton12,Ton13a,Ton13b].

Chapter 2

Definitions and Preliminaries

This chapter describes the technical background that is needed for understanding the rest of the text. First we introduce the formalism related to the complexity theory, graph theory and metric spaces. Then we present the main definitions and core framework of the Scheduling and Capacity problems, Conflict Graphs and power schemes.

2.1 The Complexity of Problems and Approximation Al- gorithms

Many optimization problems considered in this thesis are (believed to be) hard to solve.

The standard way of describing the hardness of problems is through the complexity classes. Two important classes of problems that we consider are P and N P. Below is a brief description of these complexity classes. In order to define these classes, a special form of optimization problems called decision problems is considered. Informally, a decision problem is a question with only two possible answers: “yes” or “no”.

An algorithm solving a decision problem is called apolynomial-time algorithm (or simply polynomial algorithm) if there is a polynomialp(n) such that for each inputxthe runtime of the algorithm is upper-bounded by p(|x|), where |x| is the size of the input. The complexity classP is the class of decision problems that can be solved by a polynomial- time algorithm. Informally, the complexity class N P is the class of decision problems which have the property that if a (claimed) solution for such a problem is given, then there is a polynomial-time algorithm for “checking” the correctness of that solution.

In particular, P ⊆ N P. Given two decision problems Π and Π⁰, we say that Π⁰ is polynomially reducible to Π, if there is a polynomial algorithm that, taking as input an instanceπ⁰ ∈Π⁰, outputs an instance π∈Π such that the answer to problemπ is “yes”

11

if and only if the answer to problemπ⁰ is “yes”. Such an algorithm is called areduction.

We say that two problems are polynomially equivalent if each of them is polynomially reducible to the other one. The classes of N P-complete and N P-hard problems are defined as follows.

Definition 2.1. A decision problem Π isN P-hard if and only if for each Π⁰ ∈ N P, Π⁰ is polynomially reducible to Π. An N P-hard problem is calledN P-complete if Π∈ N P.

It follows from the definition above that if there is a polynomial-time algorithm that solves an N P-complete problem, then P = N P. Nevertheless, based on decades of experience, it is widely believed that that is not the case [Gas12]. Thus, it is believed that there are no polynomial-time algorithms solvingN P-complete problems. See [GJ79]

for the formal theory of N P-completeness and a long list of N P-complete problems.

A common practice for dealing with N P-hard optimization problems¹ is to design polynomial-time algorithms that find solutions provably close to the optimum. Such an algorithm is called an approximation algorithm for the given problem. In an opti- mization problem it is required to find afeasible solution that optimizes the value of the objective function. An approximation algorithm finds feasible solutions that are proven to yield close to the optimum value of the objective function. Let Π be an optimization problem andopt(π) be the optimum value for the instance π∈Π. Then we say that an algorithm approximates Π with an approximation factor K ≥1 if for each π ∈Π, the algorithm finds a feasible solution with value at mostK·opt(π) for minimization prob- lems or finds a feasible solution with value at leastopt(π)/Kfor maximization problems.

In this thesis we will mainly deal with approximation algorithms.

2.2 Graphs

In this section we describe several notions from the theory of graphs that will be used for describing and analysing conflict graph models.

Definition 2.2. LetV be a finite set which we call thevertices and let E be a set of unordered pairs of vertices which we call edges. Then the pair G = (V, E) is called a (undirected) graph.

We assume that there are no edges of the form (a, a) (loops). Note that since we assume thatE is a set, there are also no multiple copies of the same edge (parallel edges).

1Those are the optimization problems such that expressing them as decision problems yields an N P-hard problem.

A graphG⁰ = (V⁰, E⁰) is called a subgraph of a graphG= (V, E) ifV⁰ ⊆V andE⁰ ⊆E.

G⁰ is called a spanning subgraph if G⁰ is a subgraph of G and V⁰ = V. G⁰ is called an induced subgraph of G if G⁰ is a subgraph of G and for all u, v ∈ V⁰, (u, v) ∈ E⁰ if and only if (u, v)∈E. Note that an induced subgraph is uniquely defined by a subset V⁰ ⊆V. We denote the subgraph of Gwith vertex set V⁰ byG[V⁰].

If (u, v) ∈E, we say that verticesu and v are adjacent inG. Otherwise we say that u and v areindependent. Theneighborhood of a vertex v is the set of adjacent vertices:

N_G(v) ={u: (u, v)∈E}.

The degree δ_G(v) of a vertex v is the number of vertices adjacent tov:

δ_G(v) =|N_G(v)|.

A set of verticesSis called anindependent setif for allu, v∈S,uandvare independent.

We denote byτ(G) the number of vertices in a maximum cardinality independent set of G. The maximum independent set problem is to find a maximum independent set. In themaximum weighted independent set problem, given a graph Gand weightsw(v)≥0 for the vertices of G, it is required to find an independent set that maximizes the sum of weights of its vertices.

By a (vertex) coloring of a graph G= (V, E) we will understand a partition the set of vertices of Ginto disjoint independent subsets: V =S

iU_i andU_i∩U_j =∅ for all i6=j and each U_i is an independent set. The number of such subsets is called the number of colors in the coloring. The minimum number of colors needed to color a graph G is called thechromatic index ofGand is denoted byχ(G). Thecoloring problem is to find a coloring corresponding to the chromatic index.

In general graphs, the coloring and maximum independent set problems are hard to solve. In fact, it is shown that for any fixed > 0, there is no O(n)-approximation algorithm for either of these problems, unlessP =N P [FK96,H˚as96,Zuc07]. However, these problems can be efficiently approximated for certain subfamilies of graphs such as the graphs with boundedinductive independence number which we define next.

2.2.1 Inductive Independence Number

Let G = (V, E) be a graph with |V| = n and let v1, v2, . . . , v_n be an ordering of the vertices. We denoteV_i =v_i, v_i+1, . . . , v_n.

Definition 2.3. [YB12] Ak-inductive independence ordering of vertices is an ordering v₁, v₂, . . . , v_n such that for any v_i, 1≤i≤n, τ(G[N_G(v_i)∩V_i]) ≤k. The minimum of such kis called the inductive independence number and is denoted by λ(G).

Consider Algorithm2.1for finding an independent set in a graph G.

Algorithm 2.1 Approximation of the Maximum Independent Set Problem.

1. Input: graph G= (V, E) and a k-inductive independence orderingv₁, v₂, . . . , v_n. 2. LetI =∅.

3. Consider the vertices in the following order: v1, v2, . . . , v_n. 3.1. IfI∪ {v_i} is independent,I ←I ∪ {v_i}.

4. Output: I.

Theorem 2.4. Algorithm 2.1 is a k-approximation algorithm for the maximum inde- pendent set problem for graphs Gwith λ(G)≤k.

Proof. It is easy to check that the algorithm is polynomial. Let I^∗ be a maximum independent set of G and let F(v_i) =N_G(v_i)∩V_i. Note that V =S

vi∈I(F(v_i)∪ {v_i}) because for each v_j ∈V \I there is some v_i ∈ I such that v_j ∈ F(v_i) (the reason why the algorithm did not select v_j). Since τ(F(v_i)∪ {v_i}) ≤ k, I^∗ cannot contain more than k vertices from F(v_i)∪ {v_i}, i.e. |I^∗ ∩(F(v_i)∪ {v_i})| ≤ k for all v_i ∈ I; hence,

|I^∗| ≤k|I|.

In fact, a more general theorem holds [YB12] (originally presented in [AADK00]).

Theorem 2.5. There is a polynomial algorithm that, given a k-inductive independence ordering for a graph G, finds ak-approximation for the maximum weighted independent set problem for G.

Similar ideas can be used to approximate the coloring problem. Consider Algorithm2.2 for approximating the coloring problem. Let us first state the following simple fact regarding Algorithm 2.2 which is related to an upper bound for the coloring number obtained in [SW68].

Lemma 2.6. Let v1, v2, . . . , v_n be any ordering of the vertices of a graph G. Denote F(v_i) =N_G(v_i)∩V_i. Algorithm 2.2, applied to G with the given ordering, uses at most max_i{|F(v_i)|}+ 1 colors.

Algorithm 2.2 Approximation of the Coloring Problem.

1. Input: graph G= (V, E) and a k-inductive independence orderingv1, v2, . . . , v_n. 2. LetIt=∅fort= 1,2, . . ..

3. Consider the vertices in the following order: v_n, v_n−1, . . . , v₁.

3.1. Find the first tsuch thatI_t∪ {v_i} is independent and set I_t←I_t∪ {v_i}.

4. Output: collection{I_t:I_t6=∅}.

Proof. Note that at each step, when we choose a setI_tfor a vertexv_i, there are at most

|F(v_i)|“busy” sets, wherev_i cannot be added. Hence, the number of colors used by the algorithm is at most max_i{|F(v_i)|}+ 1.

Theorem 2.7. [YB12] Algorithm 2.2 is a k-approximation algorithm for the coloring problem for graphs Gwith λ(G)≤k.

Proof. Let v_i0 be a vertex such that |F(v_i0)|= max_i{|F(v_i)|}. Recall that τ(F(v_i0)∩ {v_i0})≤k; hence, each coloring of G must use at least l_F(v

i0)+1 k

m

colors (otherwise at least one of the independent sets ofF(v_i0)∩ {v_i0}will contain more thanknodes). This, combined with Lemma 2.6, completes the proof.

Theorem 2.7 has a longer proof in [YB12]. The idea of the proof presented here is borrowed from [MBI⁺95].

2.3 Metric Spaces

We will use the notion of a metric space in order to model sets of wireless devices located in some “place”, and the distances between those devices. We model wireless devices as point objects, i.e. the sizes are ignored.

In general, a metric space is an arbitrary setM together with a functiond:M×M →R such that for any elementsa, b, c∈M, the following conditions hold:

X d(a, b)≥0,

X d(a, b) = 0 if and only ifa=b, X d(a, b) =d(b, a) (symmetry),

X d(a, b)≤d(a, c) +d(c, b) (triangle inequality).

The elements of a metric space are calledpoints, and the function dis called distance.

We can use the distance function to define the notion of a ball and a ring in a metric space.

Definition 2.8. Let M be a metric space and let a∈M. Then the (open) ballB(a, r) of radius r >0 centered at pointais the set

B(a, r) ={b∈M :d(a, b)< r},

and the ring centered at the pointaand having widthw and outer radius r > w is the set

R(a, r, w) ={b∈M :r−w≤d(a, b)< r}.

2.3.1 Doubling Metric Spaces

The best known examples of metric spaces are the Euclidean spaces R^m with the Eu- clidean distance between points. All of the results in this thesis hold in Euclidean metric spaces. However, sometimes we will need only a part of the properties of Eu- clidean spaces for proving our results. In these cases we will state the results for more general metric spaces that encapsulate some of the features of Euclidean spaces that are most relevant to our setting. These metric spaces are called doubling metric spaces.

Doubling metric spaces generalize the property of the Euclidean spaces that in a ball of radius 2r one could fit only a finite number of disjoint balls of radius r. For a more formal definition of doubling spaces we need another definition.

Assouad dimension or doubling dimension of a metric space can be defined as the infi- mum of all numbersδ >0 such that every ball of radiusr >0 has at most C^−δ points of mutual distance at leastrwhereC≥1 is a constant (independent of the ball),δ >0 and 0< ≤1. Doubling metric spaces are precisely the metric spaces of finite doubling dimension [Hei00].

It follows from the definition of doubling metric spaces that, given a metric space of doubling dimensionδ, any ball of radiusr >0 can contain (as a subset) at mostC(2)^−δ disjoint balls of radiusr.

In particular, R^m is a doubling metric space of dimension m [Hei00]. For example, on the Euclidean Plane, one can “fit” two disjoint disks of radius 1 inside a disk of radius 2, but it is not possible to fit three such disks inside a disk of radius 2 (see Fig. 2.1).

This kind of results can be proven using e.g. arguments based on areas/volumes of balls in Euclidean spaces.

Figure 2.1: Doubling property ofR².

2.3.2 Euclidean Spaces and the Lebesgue Measure

We will use the standard Lebesgue measure for measuring volumes and areas in Eu- clidean spaces. We denote the Lebesgue measure by µ. The following are general properties of measures that we will use. Let{E_i}_i∈I be a countable collection ofdisjoint measurable subsets of R^m.

X µ(E_i)≥0 (nonnegativity), X µ(∅) = 0

X µ S

i∈IE_i

=P

i∈Iµ(E_i) (countable additivity).

We will also use the following property of the Lebesgue measure: let B_r1 and B_r2 be any two balls inR^m with radii r1 and r2 respectively, then

µ(B_r1) µ(B_r2) =

r1

r₂ m

.

2.4 Capacity and Scheduling Problems

The general theoretical problems that we consider in this thesis are as follows. Given is a set Γ of n links, numbered from 1 to n: Γ = {1,2, . . . , n}. Each link represents a transmission request between a sender node and a receiver node in a wireless network.

The nodes are located in a metric space, and the sender node of each link is assigned a certain transmission power level². Then, using a specific interference model, one can define a feasible set of links, i.e. a set of links that can transmit with the given power levelsall in the same time slot without (too much) interference. In this thesis we consider

2In some settings we will consider the power level of the receiver node as well, e.g. when a bidirectional communication is considered.

Figure 2.2: A set of links.

two main optimization problems, which are stated below.

Definition 2.9. Capacity problem: Given a set of links Γ with corresponding power assignment, select a maximum cardinality feasible subset of Γ.

Definition 2.10. Scheduling problem: Given a set of links Γ with corresponding power assignment, split Γ into the minimum number of feasible subsets, each of which can be assigned to a different time slot. Each such collection of subsets is called a schedule, and the number of subsets is called the length of the schedule.

Of course, the sizes of feasible sets and schedules depend on the power assignment of the nodes and the interference model. However, there is a general algorithmic relationship between the capacity problem and the scheduling problem which does not depend on the underlying model. The following theorem was proven in [GWHW09] (the proof here is slightly different). Suppose that there is an algorithmAfor finding large feasible subsets of links. Then one can derive the followinggreedy algorithm scheme for finding feasible schedules:

Algorithm 2.3 Greedy Algorithm Scheme for Scheduling.

1. Input: set of links Γ.

2. Repeat Step 2.1 until Γ =∅.

2.1. Set Γ←Γ\S whereS is the output of algorithmA applied to Γ.

3. Output: the collection of obtained feasible subsets.

Theorem 2.11. If the approximation factor of algorithmAis inO(1), the corresponding greedy scheduling scheme approximates the scheduling problem with an approximation factor in O(logn) where n=|Γ|.

Proof. LetT denote the optimum schedule length for Γ and leta_kdenote the size of the feasible set obtained atk-th iteration. Note that at iterationk+ 1 the size of an optimal

feasible subset is at least (n−P_k

1a_t)/T. Letρ >1 denote the approximation factor of algorithm A. Then

a_k+1 ≥ n−Pk 1a_t ρT , so

k

X

1

at+ρT a_k+1≥n. (2.1) Also, by definition of A,a_t≥a_k+1/ρfort≤k. It follows that

dρ²Te

X

1

a_t≥ρT a_dρ2Te+1,

which, applied to (2.1), gives

dρ²Te

X

1

a_t≥n/2,

i.e. the first dρ²Te iterations of the algorithm halve the size of Γ. This means that the algorithm will stop after O(ρ²Tlogn) iterations. This completes the proof.

Next we define the specific models of wireless networks that we will adopt in order to obtain algorithmic results and theoretical bounds for the capacity and scheduling problems. These models describe signal propagation, interference and other phenomena present in wireless networks which play an important role in defining the capabilities and constraints of networks.

2.4.1 The Path-Loss Model for Signal Decay and Cumulative Interfer- ence

Among other kinds of distortion, wireless signal experiences path loss. Let the sender node of each linki∈Γ be assigned a power levelP(i)>0 according to some assignment policy orpower scheme (see Section2.6for details). According to thepath-loss propaga- tion model [Rap02], when the sender node of a linkitransmits with a power levelP(i), that signal reaches the receiver node of a linkj with (average) power

I_ij = P(i) d^α_ij

where d_ij = d(s_i, r_j). Here d(., .) denotes the distance function of the metric space and α > 0 is the path loss exponent (it is typically assumed that 2 < α < 6 in prac- tice [Rap02]). Of course, the formula above does not make sense for very small values of d_ij: it is valid if the receiver is in the far-field of the sender, i.e. is further than a certain distance which depends on the dimensions of the antenna and wavelength of the

signal. We assume in this thesis that all the nodes in the network instances considered are in each others far-fields so that the equation above holds.

We denote byI_ij theinterferencecaused to linkjby linki. We assume that the antennas of the wireless nodes are omni-directional, so that I_ij does not depend on the direction, but only on the distance. According to the same principle of path loss, the signal of the sender of the link i reaches the receiver node of the same link with power P_i = ^P(i)_lα

i , wherel_i =d(s_i, r_i) is the length of the linki. These values are depicted in Fig.2.3.

Figure 2.3: Path loss and interference.

Interference has the additivity property, i.e. the total interference caused by a set of links S to a link j is

I_Sj=X

i∈S

I_ij.

We make the convention I_ii= 0 for simplicity of formulas.

We assume that the interference and signal defined above are static. In reality these formulas describe the averaged power levels. However, it has been shown that at least in the case of Rayleigh fading where the power has an exponential distribution around these mean values the solutions of the scheduling problem differ by small factors from the solutions obtained in the static setting [DHK12].

2.4.2 SINR Model of Signal Reception

In the Signal to Interference and Noise Ratio (SINR) model, a transmission correspond- ing to a link i is successful in presence of a set S of concurrently transmitting links if and only if the SINR ratio is over a threshold valueβ ≥1:

P_i P

j∈SI_ji+N_i ≥β, (2.2)

where the constant N_i ≥0 denotes the environmental noise around the receiver of the link i. In other words, the signal of the link i must be at least β times greater than the total interference by other transmitting links plus the noise. Recall that we assume I_ii= 0.

Definition 2.12. In the SINR model, a subsetSof Γ is called SINR-feasible (or feasible) iff (2.2) holds for each link i∈S.

Thus, schedules and capacity can be defined in the SINR model using this defini- tion of feasibility. We denote the optimum SINR-capacity of a set Γ with respect to power scheme P as OP T C_P(Γ), and we denote the optimum SINR-schedule length by OP T SP(Γ).

2.4.3 ISR and Zero Noise

By a simple manipulation, the SINR condition takes the following form:

ISR_P(S, i) =ci

X

j∈S

I_ji P_i ≤ 1

β, (2.3)

where c_i = _P ^Pi

i−βNi. We call the left part of (2.3) Interference to Signal Ratio (ISR)³. This form of the SINR condition is widely preferred to the original form. The reason for this is that in this form the left part of the formula is more comfortable to manipulate because of additivity⁴:

ISR_P(S, i) =X

j∈S

ISR_P(j, i).

Under this notation, we call a set of links S a γ-feasible (or γ-SINR-feasible) set if for each linki∈S,

ISR_P(S, i)≤ 1 γ.

Similarly, each partitioning of Γ into a number of disjoint γ-feasible subsets is called a γ-feasible schedule. Using the additivity ofISR, [HW09] proved the following result that is very useful in asymptotic computations involving the SINR model. The theorem shows, in particular, that if one increases the β threshold of the SINR condition by a constant factor then the length of the optimum schedule changes by at most a constant factor. It should be noted that the theorem does not use the particular form of ISR, but just the fact that it is additive. This means that it is equally applicable for different

3In some of the related literature this is called “affectedness” or “affectance”.

4We will usually not write the set braces in the ISRP formula when the first argument is an one- element set. Since the letters for sets and elements are different, hopefully, no confusion will happen.

setting e.g. for the bidirectional SINR model (Chapter 8), provided that we have the additivity of ISR. We present the algorithm and the corresponding theorem below.

The algorithm uses a subroutine that we callfirstfit.

Procedurefirstfit

1. Input: a power schemeP, a numberγ⁰ >0 and a set of linksi₁, i₂, . . . in this fixed order.

2. Take empty setsS1, S2, . . . in this fixed order.

3. Fork= 1,2, . . ., do:

3.1. SetS_t←S_t∪{i_k}, wheretis the smallest index such thatISR_P(S_t, i_k)≤ _2γ¹0. 4. Output: obtained non-empty sets S1, S2, . . . in their original order.

Algorithm 2.4 [HW09] Transforming a γ-feasible schedule into a γ⁰-feasible schedule.

1. Input: aγ-feasible scheduleS₁, S₂, . . . of a set of links Γ w.r.t. power assignment P and a numberγ⁰>0.

2. SetW ← ∅. // an empty collection of sets 3. Fort= 1,2, . . ., do:

3.1. Leti1, i2, . . . , i_k be the links of S_t in this fixed order.

3.2. Set {T₁, T₂, . . . , T_p} ←firstfit(γ⁰, P, i₁, i₂, . . . , i_k).

3.3. Forf = 1,2, . . ., do:

3.3.1. Let j1, j2, . . . , j_h be the links ofT_f in the opposite of their original order inS_t.

3.3.2. Set W ←W∪firstfit(γ⁰, P, j₁, j₂, . . . , j_h).

4. Output: γ⁰-feasible scheduleW.

Theorem 2.13. [HW09] For any γ⁰ > γ > 0 and any power assignment P, Algo- rithm 2.4, applied to a γ-feasible schedule, outputs a γ⁰-feasible schedule, increasing the length of the original schedule by a factor of at most d2γ⁰/γe².

Proof. Consider the set S_t for some index t. Let i ∈ T_p be any link in the last set Tp. Since i did not “fit” in the sets T1, T2. . . , Tp−1, we have ISR_P(T_f, i) > _2γ¹0 for f = 1,2, . . . , p−1. Using additivity we getISR_P(S_t, i) > ^p−1_2γ0 . On the other hand, we have assumed thatISR_P(S_t, i)≤ _γ¹. From these inequalities we get that one application of firstfit splits the input set into p ≤ d2γ⁰/γe subsets. Since we have two nested applications of firstfit, the resulting number of sets is bounded by d2γ⁰/γe². It