Sharif Digital Repository

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... 

A social network is a social structure made up of individuals (or organizations) called ”nodes”, which are tied by one or more specific types of interdependency. These networks are new forms of distributed larg media which actually plays a critical role in distributing information and news among the people. because of the size and autonomos aspect of these kind od medias, taking control of them seems impossible. with the best hope we could think of some ways to detect information cascades through network.for example we could select a set of nodes as sensor inorder to detect information cascades as effectively as possible. in this thesis we studied cascade detection problem in scenario based... 

Problem of 2-colorability of hypergraphs was introduced by Erd?os in 1963, and there are many results about this problem until now. In this thesis we consider this problem from the algorithmic viewpoint. In all of the algorithms we assume that the given hypergraph is 2-colorable and we want to color it. By using a simple construction, it was shown that for every r = 3, the problem of coloring of hypergraphs is as hard as the problem of coloring graphs. If NP ?= ZPP12, it is impossible to approximate the chromatic number of r-uniform hyhpergraphs with n vertices by a factor of n1-? for any fixed ? > 0, in time polynomial of n. In reference [KS03] a coloring algorithm with approximation factor... 

In this thesis, we study the heterogeneous vehicles routing problem, which is a general-ization of the well-known travelling salesperson problem. In this problem, we are given a set of heterogeneous vehicles located in specific depots, along with a set of customers in the form of a graph. The vertices of the graph represent clients and depots, and the weight of edges represents the cost of travel between vertices of the graph. The goal is to find a subgraph for each vehicle so that the union of the subgraphs contain all customers and the total travel cost is minimized. The heterogeneity here means that the cost of traversing edges for each vehicle might be different. In this work, we briefly... 

This thesis is concerned with a fundamental problem in computational geometry. This problem inspects covering a space with a set of points provided that each point of the space is visible to at least one point from the selected subset. Applications of these problems are in the areas such as geographic information systems (GIS), robotics,computer graphics and the military.The investigated space is a planar rectangular region which is partitioned into smaller rectangles by connected edges. The objective of this study is to find an optimized corridor; i.e., a connected subset of edges which contains at least one point from each rectangle. For the minimum diameter corridor problem, an exact... 

In the general case of minimum color-spanning tree which is one of the color-spanning set problems, given a weighted graph with n vertices of k different colors, the goal is to find a subtree of minimum weight such that vertices of this subtree include all the colors in the graph. In the planar case, the input is a complete graph with n colored vertices on the plane and the weight of each edge is the Euclidean distance between its corresponding vertices. In this thesis we consider the problem of minimum color-spanning tree. To this end, first we present various color-spanning set problems and some other related problems like Steiner tree and we study the previous work on these problems. Then... 

In this researh, we study the problem of profit maximization using viral marketing in social networks. In our model, the individual’s ultility value consists of two parts: a private utility and a value obtained by effects of other network members. Each member selects the service that maximizes its utility and Server can exhibit services in different qualities. First we focus on the problem from the Game Theory viewpoint and discuss on the existence and optimality of equilibrium. Next, we show that finding the best equilibrium is NP-hard. For the case of two services, we show that while P ̸= NP, no approximation algorithm with factor better than 2 exists for this problem. Finally, we try to... 

Designing a rapid transit network which is capable of providing a convenient coverage for passengers is studied in thisdoctoral thesis. First, a mathematical model of rapid transit network design which is a mixed integer program is presented (C_RTND). Objective function of the model is maximization of acceptable transit coverage and the model is capable of considering different rapid transit technologies. Furthermore, for each technology capacity constraint is considered in the model. Second, a heuristic algorithm is developed and used to solve the proposed model. The algorithm consists of 3 main parts: candidate line generation, network evaluation and a search algorithm for selecting a good... 

In this research, we consider the question of finding approximate bounds on some of computational geometry problems with uncertain data. We use color spanning set to model the uncertainty. Given a set of n points colored with m colors in d-dimension. The problem of minimum diameter in color spanning set model is finding a set of m points with distinct colors so as to minimize diameter of the points. We give an algorithm with an approximation factor (1 + ϵ) with running time O(21ϵd .ϵ−2d.n3) which improves the previous results. Next,we consider a new problem of finding bounds on unit covering in color-spanning set model:Minimum color spanning-set ball covering problem is to select m points of... 

Approximate Agreement (AA) is one of the most important issues in distributed networks in which non-faulty processes execute a voting algorithm to reach the values that are very close to each other. AA for Partially Connected Networks (PCNs) is very limited because a node may not have a complete view of the whole network which makes it difficult to reach the convergence. This research proposes a convergent Fuzzy algorithm for synchronous distributed networks in the presence of faulty or Byzantine processes. When each node receives a new value from its neighbors,it runs the convergent Fuzzy algorithm. If the received value is not in the expected range of values, the processor can discard the... 

Given a graph G with a set of terminals, two weight functions c and d defined on the edge set of G, and a bound D, a popular NP-hard problem in designing networks is to find the minimum cost Steiner tree (under function c) in G, to connect all terminals in such a way that its diameter (under function d) is bounded by D. Marathe et al. [1] proposed an (O(log2n);O(log2n)) approximation algorithm for this bicriteria problem, where n is the number of terminals. The first factor reflects the approximation ratio on the diameter bound D, and the second factor indicates the cost-approximation ratio. Later, Kapoor and Sarwat [2] introduced a parameterized approximation algorithm with performance... 

recent years, by the huge increase in the number of components on a board, the bus routing problem and minimizing the number of layers of a board has emerged as a significant challenge in designing Printed Circuit Boards (PCBs). Two related theoretical problems to bus routing are the Rectangle Escape Problem (REP) and the maximum independent set of rectangles problem where both of these problems are NP-hard in the general form. In this thesis we study some variants of these problems and try to improve the running time or approximation factor of the best algorithms previously provided for solving them. The maximum disjoint set of boundary rectangles problem is a variant of the maximum... 

The problem of finding the near neighbours is as follows: given a set of npoints, build a data structure that for any query point, can quickly find all points in distancer from the query point. The problem finds applications in various areas of computer science such as data mining, pattern recognition, databases, and search engines. An important factor here is to determine the number of points to be reported. If this number is too small, the answers may be too homogeneous (similar to the query point), and therefore, convey no useful information.On the ther hand, if the number of reported points is too high, again the informativeness decreases because of the large output size. Therefore, in... 

Due to recent advancements and wide usage of location detection devices, huge amount of data are collected by GPS and cellular technologies which exhibits moving objects trajectories. Using this information, it is possible to track a set of objects over a long period of time, as happens for instance in studying animal migration. Always, in these situations it is undesirable or even impossible (due to process and storage limits) to store the complete stream of positioning data. Then, simplifying a trajectory as a data reduction technique is the option for resolving such problems. Moreover, there is an increasing interest in queries capturing ”aggregate” behavior of trajectories as groups like... 

Evaluating the enviroment for achieving efcient kwnoledge in order to justifed environment recognition and making decision based on data analysis is proponed in many felds.One of these problems is Visibility Problem. The Recieved data from real world as input, for visibility problem, is uncertain such that the recieving data of two different measurments will be different with high probability. Therefore, we defne the goal of this research to solving the visibility problem conditioned to that objects or segments are uncertain.Consider ”S” as a set of ”n” uncertain segments in plane. Each uncertain segment locates in two certain places and with probability 1 2 in each place, the goal is to... 

Planar visibility computing is defned as determining the region of the plane that is visible from a specifc observer. This concept has many applications in computer graphics, robotic and computer games. In certain visibility problems, counting the number of visible objects in an appropriate time is required. For obtaining a solution fast, current algorithms give an approximated count. In this thesis, we consider visibility testing problem and visibility counting problem.For a given set S = fs1; s2; :::; sng of non-intersecting segments and a query point p in the plane, the visibility testing problem checks the inter-visibility of p and a segment si 2 S and the visibility counting problem... 

Data clustering is simply the process of partitioning data in to non empty subsets, called clusters, such that in every clusters members are related due to an pre-defined objective function. Assume that there are n points in eulcidian d-dimensional space and the goal is to determine k points in space as a representive of each cluster so that in every cluster, members have the least value of an objective function which is related to their distances from representatives. From most important types of data clustering problem we may mention k-center, k-median and k-means problem in which the objective functions are respectively, minimizing the maximum, sum and sum of squares of distance of every... 

The Steiner forest problem is one of the most fundamental issues in network design that emerges in various contexts, such as the design of communication networks,transportation networks, VLSI circuits and phylogenetic network reconstruction. In this problem, we are given a connected weighted undirected graph and a special subset of vertices that are partitioned in pairs. The goal is to fnd a minimum-cost sub-graph in which each special vertex is connected to its mate. In this thesis we review a set of selected solutions for this problem and a few other related problems.For example, we study modeling and solving the s-t shortest path problem in order to gain insight into current Steiner... 

Nowadays, we observe the growth in the number of mobile smart phones. These smart phones have had an impressive progress in software and hardware. Many studies have focused on utilizing this new platform. One of these areas of research is collecting information from environment.Two major challenges for a successful crowdsourcing system are engaging a sufficient number of workers and validating of their registered data, which make the system more reliable. Collecting data from different points of a territory can be defined as some tasks and should be assigned to users near those tasks properly. Some users want a monetary reward for the cost they incur. On the other hand, information... 

In this thesis, we focus on a subset of geometric optimization problems (including k-center) in the Sliding Window model. The sliding window model is driven from the Data Stream model in which input points arrive one by one and the space is limited. The main diffrenece of these two models is that in the sliding window model we are interested in the N latest points not all of the arrived points. In this thesis, we study Minimum Enclosing Ball, 2-center, and Euclidean k-center in the Sliding Window model. We provide a (1 + ")-approximation algorithm for MEB in d-dimensions. To our knowledge there is no algorithm for MEB in d-dimensions where d >2. We also provide a (1 + ")-approximation...