Sharif Digital Repository

There are many ways to color the vertices and edges of graphs, such as, rainbow connection, vertex coloring and dynamic coloring. In this thesis, in Chapter 1 we introduce a new coloring, we consider its relationship with some other colorings and we investigate its computational complexity. In chapter 1, we focus on the proper orientation number. The problem of orienting the edges of a given simple graph so that the maximum indegree of vertices is minimized was introduced in 2004. We show that there is a polynomial time algorithm for determining the proper orientation number of a given 3-regular graph. But it is NP-complete to decide if the proper orientation number of a 4-regular graph is 3... 

This thesis is concerned with studying several aspects of the generalized isoperimetric problems on weighted graphs. In this regard, as a generalization of well-known Cheeger constant, we define k-th isoperimetric constant as the minimum of normalized outgoing flow over all k-subpartitions (disjoint subsets) of the vertex set. Also, we investigate basic properties of these parameters and particularly we prove a Federer-Fleming-type theorem. We then introduce a generalized Cheeger conjecture which is a generalization of celebrated Cheeger inequality and relates defined isoperimetric constants to higher eigenvalues of Laplace operator. In order to find a partial proof for this conjecture, we... 

In general, Constrain Satisfaction Problems (CSP’s) are NP-complete, however, this is not the case, if we impose some restrictions on the constraints. Due to importance and applications of CSP’s in different areas of engineering and science including artificial intelligence, relational databases and logic, investigating the computational complexity of CSP’s is an important issue. Computational complexity of CSP’s has been investigated from a variety of aspects, including the Universal Algebraic approach that has given rise to a solution of dichotomy conjecture recently. In this thesis based on contributions of Bulatov, Krokhin and Jeavons, we try to explain this universal algebraic approach... 

Computational complexity of graph problems is an important branch in the-oretical computer science. We introduce to some of ideas for computing the complexity of graph problems with some kind and beautiful examples. Next, we show hardness and inapproximability of some problems. Representation number of graphs has been introduce by Pavel Erdos by Number theory. We prove n1−ϵ inapproximability of that. Lucky number η has been studied by Grytczuk et.al . We show for planar and 3-colorable graphs, it is NP-Complete whether η = 2. Note that since a conjecture, for those graphs, 2 ≤ η ≤ 3. Also for each k ≥ 2, we show NP-completeness of η ≤ k for the graphs. Proper orientation number −→ is a... 

The concept of defining set in matching theory, recently has been taken into consideration by chemists and mathematicians because of several important applied problems in chemistry and mathematics. This concept is studied extensively in vari-ous areas of combinatorics and graph theory. The idea is to obtain total combinatorial structure of one object uniquely, based on some special information about it. Defin-ing set has been studied in various areas of graph theory like colouring, matching , orientations of graphs and etc. and many researches have been done in these areas. What we focus on, in this thesis is the defining set in matching theory that has been called “forcing set”. A minimum... 

Scheduling algorithms are responsible for allocation of various types of resources, interference management, and choosing the times and paths through which messages are to be delivered in a communication network. We consider certain fundamental problems in network scheduling. Any scheduling algorithm, to be practical, must have tractable computational complexity and tolerable delay. In the first part of the thesis, we address the following question: “for a given class of networks, does there exist a scheduling algorithm that performs polynomially many computations (in terms of the network dimensions), while keeping the expected delay of the network polynomially bounded (again in terms of the... 

Robust model predictive control(RMPC) is a control strategy that has been widely adopted in industry and academic researches.In this project we have modified robust nonlinear model predictive control using SOS and dynamic feedback. In this control strategy we will linearize the nonlinear model of the system about it’s operating point. We will consider the error of linearization as an uncertainty and find an upper bound for uncertainty. Then we will change control objective to minimizing this upper bound. The most advantage of changing the control objective is that nonlinear system results a non-convex optimization problem but this strategy terminates a convex optimization problem. To find... 

Ising model. It means that the partition function of each lattice model is equal to the partition function of 2D Ising model. This completeness is proved by using concepts and techniques from quantum information theory and is based on the universality of cluster states. We have now proved this important result, by using independent and general concepts and methods which are accessible to a wide audience of researchers across many disciplines. Furthermore, our method has the advantage of eing algorithmic in nature so that, by following a set of simple graphical transformations, one is able to transform any discrete lattice model to an Ising model defined on a (polynomially) larger 2D... 

Secure multi-party computation (MPC) enables a group of mutually distrustful parties to compute a joint and agreed upon function of their private inputs without disclosing anything but the corresponding output. One of the most important secure computation protocols is private set intersection (PSI). In PSI, often two or several parties wish to find the intersection of their sets without revealing other non-common elements. There exist some other variants of PSI protocol like PSI cardinality or threshold PSI which in the former only the cardinality of the intersection set is revealed and in the latter the intersection set is revealed if its cardinality is greater (less) than a certain value.... 

Localization of radio sources has various applications in civil and military communications. Some of important localization methods consist of Angle of Arrival (AoA) estimation of the illuminated signals from the interested sources. Between various Direction Finding (DF) methods, subspace based methods are of special importance, because of their high resolution and accuracy. And from beginning of invention, because of new challenges, have been focus of numerous researches. In this thesis, first we explain the most important subspace based DF methods called MUSIC and root-MUSIC. Then a novel modification for root-MUSIC is introduced. In our new method the computational complexity of... 

Descriptive complexity refers to the difficulty of expressing a problem in a formal language, allowing for a complete and precise description of the problem, including output specifications for function problems. Despite the success of the descriptive approach in computational complexity, this approach has been predominantly used for decision problems. In contrast to decision problems, a counting problem can be considered as a function with the range of natural numbers. This thesis focuses on the examination of counting problems from a descriptive perspective, exploring the connection between their descriptive and computational versions. Toward this goal, weighted logic is utilized to... 

In this thesis, we address the problem of joint Range, Angle and Doppler frequency estimation in FDA-MIMO radars. Target parameter estimation is among the most important areas of research in radar signal processing. In this field of research, there are numerous challenges to overcome, including low estimation accuracy, a scarcity of snapshots, and the high computational complexity of algorithms. For this reason, diverse methods have been developed to solve this problem. Each of these methods has its own features and capabilities, focusing on one or more of the existing challenges. The application of these methods also depends on various factors, including the signal structure, which is... 

The subject of Secret Sharing refers to the methods that a secret information is shared (by a dealer) among a number of parties, so called participants, each having one or more shares, such that the predefined subsets of the participants, called authorized subsets, are able to regenerate the secret using their shares. A more general scheme which is called Multi Stage Secret Sharing (MSSS) is to share multiple secrets among the participants such that the authorized subsets can reconstruct the secrets in the consecutive stages; in these schemes, in addition to the individual shares, the dealer should provide the participants with a number of public values regarding the secrets. A Multi-Secret... 

The relation between BQP and PH has been one of the most fundamental open questions since the beginning of the field of quantum computational complexity. Despite the common belief that BQP contains problems outside the polynomial hierarchy, no real progress had been made in solving the problem even in relativized world. In 2009, Aaronson took the first serious step by proposing an oracle problem to separate the relation version of these two complexity classes. Finally, based on Aaronson’s works, in 2018 Raz and Tal separated BQP and PH relative to an oracle in a breakthrough work. This thesis studies the tools and techniques used to solve this challenging problem