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”
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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.