Minimizing Interference in Ultra-Dense Femtocell Networks Using Graph-Based Frequency Reuse Technique

This paper investigates the performance of graph colouring schemes for frequency assignment in Long-Term Evolution (LTE) networks with ultra-dense femtocells. The aim of the study is to minimize interference in such networks while ensuring efficient spectrum use for these femtocells. The three schemes investigated are the conventional greedy graph colouring algorithm, the saturation degree algorithm and our proposed graph-based theory (GBT) algorithm. The process of frequency assignment is similar in the last two except that the proposed GBT partitions the femtocells into independent sets for an efficient frequency re-use. The performance of these three schemes was analyzed through extensive simulations to determine the SINR and network capacity that can be obtained with the deployment of these schemes using the ITU-R P1238-7 path loss model. The outcome of this study showed that with the absence of a dynamic frequency assignment scheme, interference level is increased as the number of femtocell users within a particular coverage is increased, leading to a reduction in the capacity of such networks. Simulation results showed that all three algorithms considered have the ability to allocate frequencies to femtocells and minimize interference in a densely deployed environment, thereby increasing network capacity. The proposed GBT assigned the least sub-band thereby ensuring spectral efficiency while minimizing harmful interference. Results show that the greedy algorithm has a disadvantage of inefficiently assigning sub-bands randomly, while the saturation degree assigns more sub-bands when compared with the GBT scheme.Keywords — Femtocell, graph colouring, frequency assignment, LTE

A wide-ranging computational comparison of high-performance graph colouring algorithms

This paper reviews the current state of the literature surrounding methods for the general graph colouring problem and presents a broad comparison of six high-performance algorithms, each belonging to one of the main algorithmic schemes identified. Unlike many previous computational studies in graph colouring, a large range of both artificially generated and real-world graphs are considered, culminating in over 40,000 individual trials that have consumed more than a decade of computation time in total. The picture painted by the comparison is complex, with each method outperforming all others on at least one occasion; however, general patterns are also observed, particularly with regards to the advantages of hybridising local-search techniques with global-based operators

A study on exponential-size neighborhoods for the bin packing problem with conflicts

We propose an iterated local search based on several classes of local and large neighborhoods for the bin packing problem with conflicts. This problem, which combines the characteristics of both bin packing and vertex coloring, arises in various application contexts such as logistics and transportation, timetabling, and resource allocation for cloud computing. We introduce $O(1)$ evaluation procedures for classical local-search moves, polynomial variants of ejection chains and assignment neighborhoods, an adaptive set covering-based neighborhood, and finally a controlled use of 0-cost moves to further diversify the search. The overall method produces solutions of good quality on the classical benchmark instances and scales very well with an increase of problem size. Extensive computational experiments are conducted to measure the respective contribution of each proposed neighborhood. In particular, the 0-cost moves and the large neighborhood based on set covering contribute very significantly to the search. Several research perspectives are open in relation to possible hybridizations with other state-of-the-art mathematical programming heuristics for this problem.Comment: 26 pages, 8 figure

Quantum State Tomography with Observable Commutation Graphs

The measurement of quantum states has been a widely studied problem ever since the discovery of quantum mechanics. In general, we can only measure a quantum state once as the measurement itself alters the state and, consequently, we lose information about the original state of the system in the process. Furthermore, this single measurement cannot uncover every detail about the system's state and thus, we get only a limited description of the system. However, there are physical processes, e.g., a quantum circuit, which can be expected to create the same state over and over again. This allows us to measure multiple identical copies of the same system in order to gain a fuller characterization of the state. This process of diagnosing a quantum state through measurements is known as quantum state tomography. However, even if we are able to create identical copies of the same system, it is often preferable to keep the number of needed copies as low as possible. In this thesis, we will propose a method of optimising the measurements in this regard. The full description of the state requires determining multiple different observables of the system. These observables can be measured from the same copy of the system only if they commute with each other. As the commutation relation is not transitive, it is often quite complicated to find the best way to match the observables with each other according to these commutation relations. This can be quite handily illustrated with graphs. Moreover, the best way to divide the observables into commuting sets can then be reduced to a well-known graph theoretical problem called graph colouring. Measuring the observables with acceptable accuracy also requires measuring each observable multiple times. This information can also be included in the graph colouring approach by using a generalisation called multicolouring. Our results show that this multicolouring approach can offer significant improvements in the number of needed copies when compared to some other known methods

On the application of graph colouring techniques in round-robin sports scheduling

The purpose of this paper is twofold. First, it explores the issue of producing valid, compact round-robin sports schedules by considering the problem as one of graph colouring. Using this model, which can also be extended to incorporate additional constraints, the difficulty of such problems is then gauged by considering the performance of a number of different graph colouring algorithms. Second, neighbourhood operators are then proposed that can be derived from the underlying graph colouring model and, in an example application, we show how these operators can be used in conjunction with multi-objective optimisation techniques to produce high-quality solutions to a real-world sports league scheduling problem encountered at the Welsh Rugby Union in Cardiff, Wales

Large Cuts with Local Algorithms on Triangle-Free Graphs

We study the problem of finding large cuts in $d$-regular triangle-free graphs. In prior work, Shearer (1992) gives a randomised algorithm that finds a cut of expected size $(1/2 + 0.177/\sqrt{d})m$, where $m$ is the number of edges. We give a simpler algorithm that does much better: it finds a cut of expected size $(1/2 + 0.28125/\sqrt{d})m$. As a corollary, this shows that in any $d$-regular triangle-free graph there exists a cut of at least this size. Our algorithm can be interpreted as a very efficient randomised distributed algorithm: each node needs to produce only one random bit, and the algorithm runs in one synchronous communication round. This work is also a case study of applying computational techniques in the design of distributed algorithms: our algorithm was designed by a computer program that searched for optimal algorithms for small values of $d$.Comment: 1+17 pages, 8 figure