Some combinational optimization problems on radio network communication and machine scheduling

The combinatorial optimization problems coming from two areas are studied in this dissertation: network communication and machine scheduling. In the network communication area, the complexity of distributed broadcasting and distributed gossiping is studied in the setting of random networks. Two different models are considered: one is random geometric networks, the main model used to study properties of sensor and ad-hoc networks, where ri points are randomly placed in a unit square and two points are connected by an edge if they are at most a certain fixed distance r from each other. The other model is the so-called line-of-sight networks, a new network model introduced recently by Frieze et al. (SODA\u2707). The nodes in this model are randomly placed (with probability p) on an n x n grid and a node can communicate with all the nodes that are in at most a certain fixed distance r and which are in the same row or column. It can be shown that in many scenarios of both models, the random structure of these networks makes it possible to perform distributed gossiping in asymptotically optimal time 0(D), where D is the diameter of the network. The simulation results show that most algorithms especially the randomized algorithm works very fast in practice. In the scheduling area, the first problem is online scheduling a set of equal processing time tasks with precedence constraints so as to minimize the makespan. It can be shown that Hu \u27s algorithm yields an asymptotic competitive ratio of 3/2 for intree precedence constraints and an asymptotic competitive ratio of 1 for outtree precedences, and Coffinan-Graham algorithm yields an asymptotic competitive ratio of 1 for arbitrary precedence constraints and two machines.The second scheduling problem is the integrated production and delivery scheduling with disjoint windows. In this problem, each job is associated with a time window, and a profit. A job must be finished within its time window to get the profit. The objective is to pick a set ofjobs and schedule them to get the maximum total profit. For a single machine and unit profit, an optimal algorithm is proposed. For a single machine and arbitrary profit, a fully polynomial time approximation scheme(FPTAS) is proposed. These algorithms can be extended to multiple machines with approximation ratio less than e/(e - 1). The third scheduling problem studied in this dissertation is the preemptive scheduling algorithms with nested and inclusive processing set restrictions. The objective is to minimize the makespan of the schedule. It can be shown that there is no optimal online algorithm even for the case of inclusive processing set. Then a linear time optimal algorithm is given for the case of nested processing set, where all jobs are available for processing at time t = 0. A more complicated algorithm with running time 0(n log ri) is given that produces not only optimal but also maximal schedules. When jobs have different release times, an optimal algorithm is given for the nested case and a faster optimal algorithm is given for the inclusive processing set case

Independent sets in Line of Sight networks

Line of Sight (LoS) networks provide a model of wireless communication which incor-porates visibility constraints. Vertices of such networks can be embedded onto the cube{(x1,x2,...,xd):xi∈{1,...,n},1≤i≤d}so that two vertices are adjacent if and onlyif their images lay on a line parallel to one of the cube edges and their distance is lessthan a given range parameterω. In this paper we study large independent sets in LoSnetworks.Weprovethatthecomputationalproblemoffindingamaximumindependentset can be solved optimally in polynomial time for one dimensional LoS networks.However, ford≥2, the (decision version of) the problem becomes NP-complete for anyfixedω≥3. In addition, we show that the problem is APX-hard whenω=nford≥3.On the positive side, we show that LoS networks generalize chordal graphs. This impliesthat there exists a simpled-approximation algorithm for the maximum independent setproblem in LoS networks. Finally, we describe a polynomial time approximation schemefor the maximum independent set problem in LoS networks for the case whenωis aconstantandpresentanimprovedheuristicalgorithmfortheprobleminthecaseω=

Communication problems in random line-of-sight ad-hoc radio networks

The line-of-sight networks is a network model introduced recently by Frieze et al. It considers wireless networks in which the underlying environment has a large number of obstacles and the communication can only take place between objects that are close in space and are in the line of sight to one another. To capture the main properties of this model, Frieze et al. proposed a new random networks model in which nodes are randomly placed on an n x n grid and a node can communicate with all the nodes that are in at most a certain fixed distance r and which are in the same row or column. Frieze et al. concentrated their study on basic structural properties of the random line-of-sight networks and in this paper we focus on their communication aspects in the scenario of ad-hoc radio communication networks. We present efficient algorithms for two fundamental communication problems of broadcasting and gossiping in the classical ad-hoc radio communication model adjusted to random line-of-sight networks

Independent Sets in Line of Sight Networks

In this thesis we study the maximum independent set problem in both 2 and higher dimensional line of sight networks. The maximum independent set problem seeks to find a largest set of pairwise disjoint vertices and we will study both the decision version and the optimisation version of the problem in this thesis. The line of sight network model was introduced to provide a model of geometric graph that incorporates both range and line of sight restrictions. A LoS network model is governed by three parameters: n-size of the underlying integer grid, d-dimension of the underlying integer grid and ω the range parameter that governs how large the communication range of each vertex is, which can range from 1 to n. We first analyse the computational complexity of the maximum independent set problem for varying classes of line of sight networks governed by the dimension and range parameters d and ω. In particular, we are interested in the cases where d ≥ 2 and ω is sublinear in the size of the integer grid and where d ≥ 3 and ω is equal to the size of the integer grid, thus maximising the communication range. This naturally leads us to the design of a number of approximation algorithms for various classes of line of sight networks where the maximum independent set problem is NP-hard. Finally we study the maximum independent set problem in a restricted 2-dimensional line of sight network model. In this model, we show that the maximum independent set problem has a connection to a scheduling application. We show how methods that we develop for solving the maximum independent set problem, can also be used to solve the scheduling problem in both an offline and semi-online setting