Frequency Assignment Problem Research Articles

<i>L</i>(<i>d</i>,1)-labellings of generalised Petersen graphs

An interesting graph distance constrained labelling problem can model the frequency channel assignment problem as well as code assignment in computer networks. The frequency assignment problem asks for assigning frequencies to transmitters in a broadcasting network with the aim of avoiding undesired interference. One of the graph theoretical models of The frequency assignment problem is the concept of distance constrained labelling of graphs. Let u and v be vertices of a graph G = (V (G),E(G)) and d(u, v) be the distance between u and v in G. For an integer d ≥ 0, an L(d, 1)-labelling of G is a function f : V (G) → {0, 1, · · · } such that for every u, v ϵ V (G), |f(u) – f(v)| ≥ d if d(u, v) = 1 and |f(u) – f(v)| ≥ 1 if d(u, v) = 2. The span of f is the difference between the largest and the smallest numbers in f(V (G)). The λd,1-number of G is the minimum span over all L(d, 1)-labellings of G. For natural numbers n and k, where n > 2k, a generalised Petersen graph P(n, k) is obtained by letting its vertex set be {u1, u2, · · · , un} υ {v1, v2, · · · , vn} and its edge set be the union of uiui+1, uivi, vivi+k over 1 ≤ i ≤ n, where subscripts are reduced modulo n. In this paper, we show the λd,1-numbers of the generalised Petersen graphs P(n, k) for n ≥ 5.

<i>L</i>(<i>d</i>,1)-labellings of generalised Petersen graphs

An interesting graph distance constrained labelling problem can model the frequency channel assignment problem as well as code assignment in computer networks. The frequency assignment problem asks for assigning frequencies to transmitters in a broadcasting network with the aim of avoiding undesired interference. One of the graph theoretical models of The frequency assignment problem is the concept of distance constrained labelling of graphs. Let u and v be vertices of a graph G = (V (G),E(G)) and d(u, v) be the distance between u and v in G. For an integer d ≥ 0, an L(d, 1)-labelling of G is a function f : V (G) → {0, 1, · · · } such that for every u, v ϵ V (G), |f(u) – f(v)| ≥ d if d(u, v) = 1 and |f(u) – f(v)| ≥ 1 if d(u, v) = 2. The span of f is the difference between the largest and the smallest numbers in f(V (G)). The λd,1-number of G is the minimum span over all L(d, 1)-labellings of G. For natural numbers n and k, where n > 2k, a generalised Petersen graph P(n, k) is obtained by letting its vertex set be {u1, u2, · · · , un} υ {v1, v2, · · · , vn} and its edge set be the union of uiui+1, uivi, vivi+k over 1 ≤ i ≤ n, where subscripts are reduced modulo n. In this paper, we show the λd,1-numbers of the generalised Petersen graphs P(n, k) for n ≥ 5.

An Optimization Heuristic Based on Non-Dominated Sorting and Tabu Search for the Fixed Spectrum Frequency Assignment Problem

Frequency assignment for minimum interference is a fundamental problem in telecommunications networks. Combinatorial optimization heuristics are among the successful techniques employed to solve this problem. The heuristic proposed in this paper is a hybrid of the non-dominated sorting genetic algorithm-II and tabu search (TS) heuristics. Although several hybrid heuristics exist, in addition to the parameters of the metaheuristics, they also contain problem-specific parameters. The proposed heuristic does not have any problem-specific parameter. In each iteration, we apply TS to each element of the population to replace it with the best solution in its neighborhood. We select the elements for the population of the next generation using the principles of the non-dominated sorting. The non-dominated sorting considers two attributes of a solution: 1) interference (i.e., a measure of the violation of constraints) and 2) entropy (i.e., a measure of the diversity of the solutions in a population). Experimental results show that the proposed heuristic can produce solutions of quality better than four existing known heuristics. The gain of the proposed heuristic over a recently proposed hybrid heuristic is verified using the Wilcoxon statistical test. The convergence time of the proposed heuristic is also comparable with the existing heuristics.

Frequency assignment problem with net filter discrimination constraints

Managing radio spectrum resources is a crucial issue. The frequency assignment problem (FAP) basically aims to allocate limited number of frequencies to communication links in an efficient manner. Geographically close links, however, cause interference, which complicates the assignment, imposing frequency separation constraints. The FAP is closely related to the graph-coloring problem and it is an NP-hard problem. In this paper, we propose to incorporate the randomization into greedy and fast heuristics. As far as being implemented, the proposed algorithms are quite straight forward and are unencumbered by system parameters that need to be tuned whenever the system changes. The proposed algorithms significantly improve, quickly and effectively, the solutions obtained by greedy algorithms in terms of the number of assigned frequencies and the range. The enhanced versions of the proposed algorithms perform close to the lower bounds while running in a reasonable duration. Another novelty of our study is its consideration of the net filter discrimination effects in the communication model. Performance analysis is done by synthetic and measured data, where the measurement data includes the effect of the real 3-dimensional (3D) geographical features in the Daejeon region in Korea. In both cases, we observe a significant improvement by employing randomized heuristics.

Improving Diversity in Evolutionary Algorithms: New Best Solutions for Frequency Assignment

Metaheuristics have yielded very promising results for the frequency assignment problem (FAP). However, the results obtainable using currently published methods are far from ideal in complex, large-scale instances. This paper applies and extends some of the most recent advances in evolutionary algorithms to two common variants of the FAP, and shows how, in traditional techniques, two common issues affect their performance: 1) premature convergence and 2) the way in which neutral networks are handled. A recent replacement-based diversity management strategy is successfully applied to alleviate the premature convergence drawback. Additionally, by properly defining a distance metric, the performance in the presence of neutrality can also be greatly improved. The replacement strategy combines the principle of transforming a single-objective problem into a multiobjective one by considering diversity as an additional objective, with the idea of adapting the balance induced between exploration and exploitation to the requirements of the different optimization stages. Tests with 44 publicly available instances yield very competitive results. New best-known frequency plans were generated for 11 instances, whereas in the remaining ones the best-known solutions were replicated. Comparisons with a large number of strategies designed to delay convergence of the population clearly show the advantages of our novel proposals.

A Grassmann Matrix Approach for the Computation of Degenerate Solutions for Output Feedback Laws

The paper is concerned with the improvement of the overall sensitivity properties of a method to design feedback laws for multivariable linear systems which can be applied to the whole family of determinantal type frequency assignment problems, expressed by a unified description, the so-called Determinantal Assignment Problem (DAP). By using the exterior algebra/algebraic geometry framework, DAP is reduced to a linear problem (zero assignment of polynomial combinants) and a standard problem of multilinear algebra (decomposability of multivectors) which is characterized by the set of Quadratic Plücker Relations (QPR) that define the Grassmann variety of P. This design method is based on the notion of degenerate compensator, which are the solutions that indicate the boundaries of the control design and they provide the means for linearising asymptotically the nonlinear nature of the problems and hence are used as the starting points to generate linearized feedback laws. A new algorithmic approach is introduced for the computation and the selection of degenerate solutions (decomposable vectors) which allows the computation of static and dynamic feedback laws with reduced sensitivity (and hence more robust solutions). This approach is based on alternative, linear algebra type criterion for decomposability of multivectors to that defined by the QPRs, in terms of the properties of structured matrices, referred to as Grassmann Matrices. The overall problem is transformed to a nonlinear maximization problem where the objective function is expressed via the Grassmann Matrices and the first order conditions for optimality are reduced to a nonlinear eigenvalue-eigenvector problem. Hence, an iterative method similar to the power method for finding the largest modulus eigenvalue and the corresponding eigenvector is proposed as a solution for the above problem.

On relationship between Hamiltonian path and holes in [formula omitted]-coloring of minimum span

The L(2,1)-coloring and L(3,2,1)-coloring on a finite simple connected graph G originate from frequency assignment problem. Any L(2,1)-coloring on G of minimum possible span is a λ2,1-coloring of G and its span is the λ2,1-number of G, denoted by λ2,1(G). Any integer i, 0<i<λ2,1(G), which is left unused as a color in a λ2,1-coloring L, is a hole in L. The λ3,2,1-number of G, λ3,2,1-coloring of G and holes in a λ3,2,1-coloring are defined in a similar manner. In their paper Georges et al. (1994) referred a hole i as a gap if |Li−1|=1=|Li+1|, Lj being the set of vertices colored j by L, and the vertices in Li−1∪Li+1 are adjacent. They showed if G admits a λ2,1-coloring without any gap, then the complement Gc has a Hamiltonian path. We first explore similar phenomenon in the perspective of L(3,2,1)-coloring, a natural generalization of L(2,1)-coloring problem. Towards this, we coin a term 2-gap. Formally, a λ3,2,1-coloring Lˆ on G is said to have a 2-gap {i+1,i+2} if Lˆ has two consecutive holes i+1, i+2 and |Lˆi|=1=|Lˆi+3|, where Lˆj is the set of vertices colored j by Lˆ. We prove if G admits a λ3,2,1-coloring without any 2-gap, then Gc has a Hamiltonian path. But investigating the converse of this result poses an intriguing combinatorial problem. However, after exploring various interesting features of a 2-gap, we give a class of finite simple connected graphs for which the converse is true. For the same class of graphs, a characterization of the L(3,2,1)-coloring problem is given as an application.

Formula omitted]-Labeling of Kneser graphs and coloring squares of Kneser graphs

The frequency assignment problem is to assign a frequency to each radio transmitter so that transmitters are assigned frequencies with allowed separations. Motivated by a variation of the frequency assignment problem, the L(2,1)-labeling problem was put forward. An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1 and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number λ(G) of G is the smallest number k such that G has a L(2,1)-labeling with max{f(v):v∈V(G)}=k. Griggs and Yeh (1992) conjectured that λ(G)≤Δ2 for any graph with maximum degree Δ≥2. The Kneser graphK(a,b) is defined as the graph whose vertices correspond to all b-subsets of the a-set A={1,2,…,a}, with edges joining pairs of vertices that correspond to non-overlapping b-subsets. The chromatic number of the Kneser graph K(a,b) is a−2b+2. Füredi put forward an open problem: What is the value for the chromatic number of the square of any Kneser graph K(a,b) (the square of a graph is the graph obtained by adding edges joining vertices at distance 2)? In this article, the L(2,1)-labeling numbers of Kneser graphs K(a,b) are considered. The combined upper bounds for the L(2,1)-labeling numbers of Kneser graphs are derived by using two approaches. The exact L(2,1)-labeling number of Kneser graph K(a,b) for a≥3b−1 is obtained, and it is proved that Griggs and Yeh’s conjecture holds for Kneser graphs and the L(2,1)-labeling numbers of Kneser graphs are much better than Δ2 in most cases. We also provide bounds for the chromatic number of the square of any Kneser graph K(a,b) using the proof for the upper bounds of the L(2,1)-labeling numbers of Kneser graphs.

Upper Bound for the Radio Number of Some Families of Sunlet Graph

Objectives: In this study, Radio Coloring is used to color the graphs. The objective of this article is to analyze the bounds of Line, Middle, Total and Central graphs of Sunlet graph. Methods/Analysis: Combinatorics is an important part of discrete mathematics that solves counting problems without actually enumerating all possible cases. Combinatorics has wide applications in Computer Science, especially in coding theory, analysis of algorithms and others. An equation that expresses an, the general term of the sequence {an} is called a recurrence relation. Using the generating function of a sequence and few coloring techniques we prove the results. Findings: The Problem of finding radio coloring with small or optimal k arises in the concept of radio frequency assignment. The radio chromatic score rs(G)17 of a radio coloring is the number of used colors. The number of colors used in a radio coloring with the minimum score is the radio chromatic rn(G) of G. The radio chromatic number of Sunlet graph Sn is 10 4 if n=3i, i=1,2,... 5 if n=3i+1, i=1,2,... 6 if n=3i+2, i=1,2,... and is improved to the radio chromatic number of Sunlet graph Sn is 5 if n is congruent to 1mod 3 4 if otherwise In this paper we improve the radio chromatic number of Sunlet graph Sn and obtain the radio number of Line, Middle, Total and Central graphs of Sunlet graph. Radio Coloring has wide range of significance because radio coloring has its applications in communication theory. The paper contributes the researches in the field of computer science and combinatorics. Applications: Radio coloring is of great significance because the frequency assignment problem is modeled as a graph coloring problem assuming transmitters as vertices and interference as adjacencies between two vertices.

Frequency assignment problem in networks with limited spectrum

The frequency assignment problem (FAP) asks for assigning frequencies (channels) in a wireless network from the available radio spectrum to the transceivers of the network. One of the graph theoretical models of FAP is the L(3, 2, 1)-labeling of a graph, which is an abstraction of assigning integer frequencies to radio transceivers such that (i) transceivers that are one unit of distance apart receive frequencies that differ by at least three, (ii) transceivers that are two units of distance apart receive frequencies that differ by at least two, and (iii) transceivers that are three units of distance apart receive frequencies that differ by at least one. The relaxation of the L(3, 2, 1)-labeling called the (s, t, r)-relaxed k-L(3, 2, 1)-labeling is proposed in this paper. This concept is a generalization of the (s, t)-relaxed k-L(2, 1)-labeling (Lin in J Comb Optim 2016, doi:10.1007/s10878-014-9746-9). Basic properties of (s, t, r)-relaxed k-L(3, 2, 1)-labeling are discussed and optimal (s, t, r)-relaxed k-L(3, 2, 1)-labelings for paths and some cycles as well as for the hexagonal lattice and the square lattice are determined.

군용 통신망에서의 주파수 및 편파 지정 알고리즘 튜토리얼

본 논문에서는 군용 통신망에 대한 주파수 지정 문제를 그래프 이론 관점에서 접근하여 해결하는 방법을 소개한다. 선행 연구에서 제안한 알고리즘을 본 문제에 맞게 구현하였으며, 새로운 알고리즘을 제안하여 얻은 해의 성능이 최적의 값에 거의 근접함을 증명하였다. 또한, 이중 편파 안테나의 편파를 동시에 배정하여 주파수 지정 문제 해의 성능을 극대화하는 알고리즘을 소개하였다. In this paper we introduce a graph theory approach to solve the frequency assignment problem(FAP) for military communication network. Prior algorithms are implemented adaptively to the problem, and enhanced algorithms are proposed to show that their results approximately approached the optimum performance. We also proposed polarization assignment algorithms to enhance the FAP performances.

Joint assignment of frequency and polarization to minimize the chromatic number

Abstract In this letter, we propose graph theory-based heuristics for jointly assigning frequency and antenna polarization in a cluster-based tactical communication environment (e.g., military communications). In addition to considering the frequency assignment problem (which is an NP-hard problem), we also consider the polarization assignment issue. A practical antenna and a 3D ray tracing-based simulator are exploited to measure the interference. We show that the suggested heuristics are nearly optimal in terms of chromatics and their low complexity makes them suitable for practical usage.

Adaptive genetic algorithms guided by decomposition for PCSPs: application to frequency assignment problems

This paper proposes Adaptive Genetic Algorithms Guided by structural knowledges coming from decomposition methods, for solving PCSPs. The family of algorithms called AGAGD_x_y is designed to be doubly generic, meaning that any decomposition method and different heuristics for the genetic operators can be considered. To validate the approach, the decomposition algorithm due to Newman was used and several crossover operators based on structural knowledge such as the cluster, separator and the cut were tested. The experimental results obtained on the most challenging Minimum Interference-FAP problems of CALMA instances are very promising and lead to interesting perspectives to be explored in the future.

Distance-constrained grid colouring

Abstract Distance-constrained colouring is a mathematical model of the frequency assignment problem. This colouring can be treated as an optimization problem so we can use the toolbar of the optimization to solve concrete problems. In this paper, we show performance of distance-constrained grid colouring for two methods which are good in map colouring.

Models and Solution Techniques for Frequency Assignment Problems

Wireless communication is used in many different situations such as mobile telephony, radio and TV broadcasting, satellite communication, and military operations. In each of these situations a frequency assignment problem arises with application specific characteristics. Researchers have developed different modeling ideas for each of the features of the problem, such as the handling of interference among radio signals, the availability of frequencies, and the optimization criterion. This survey gives an overview of the models and methods that the literature provides on the topic. We present a broad description of the practical settings in which frequency assignment is applied. We also present a classification of the different models and formulations described in the literature, such that the common features of the models are emphasized. The solution methods are divided in two parts. Optimization and lower bounding techniques on the one hand, and heuristic search techniques on the other hand. The literature is classified according to the used methods. Again, we emphasize the common features, used in the different papers. The quality of the solution methods is compared, whenever possible, on publicly available benchmark instances.

Integrating Tabu Search in Particle Swarm Optimization for the frequency assignment problem

In this paper, we address one of the issues in the frequency assignment problem for cellular mobile networks in which we intend to minimize the interference levels when assigning frequencies from a limited frequency spectrum. In order to satisfy the increasing demand in such cellular mobile networks, we use a hybrid approach consisting of a Particle Swarm Optimization (PSO) combined with a Tabu Search (TS) algorithm. This approach takes both advantages of PSO efficiency in global optimization and TS in avoiding the premature convergence that would lead PSO to stagnate in a local minimum. Moreover, we propose a new efficient, simple, and inexpensive model for storing and evaluating solution's assignment. The purpose of this model reduces the solution's storage volume as well as the computations required to evaluate these solutions in comparison with the classical model. Our simulation results on the most known benchmarking instances prove the effectiveness of our proposed algorithm in comparison with previous related works in terms of convergence rate, the number of iterations, the solution storage volume and the running time required to converge to the optimal solution.

A hybrid approach based on stochastic competitive Hopfield neural network and efficient genetic algorithm for frequency assignment problem

This paper presents a hybrid efficient genetic algorithm (EGA) for the stochastic competitive Hopfield (SCH) neural network, which is named SCH–EGA. This approach aims to tackle the frequency assignment problem (FAP). The objective of the FAP in satellite communication system is to minimize the co-channel interference between satellite communication systems by rearranging the frequency assignment so that they can accommodate increasing demands. Our hybrid algorithm involves a stochastic competitive Hopfield neural network (SCHNN) which manages the problem constraints, when a genetic algorithm searches for high quality solutions with the minimum possible cost. Our hybrid algorithm, reflecting a special type of algorithm hybrid thought, owns good adaptability which cannot only deal with the FAP, but also cope with other problems including the clustering, classification, and the maximum clique problem, etc. In this paper, we first propose five optimal strategies to build an efficient genetic algorithm. Then we explore three hybridizations between SCHNN and EGA to discover the best hybrid algorithm. We believe that the comparison can also be helpful for hybridizations between neural networks and other evolutionary algorithms such as the particle swarm optimization algorithm, the artificial bee colony algorithm, etc. In the experiments, our hybrid algorithm obtains better or comparable performance than other algorithms on 5 benchmark problems and 12 large problems randomly generated. Finally, we show that our hybrid algorithm can obtain good results with a small size population.

Energy-Efficient Resource Allocation for Fractional Frequency Reuse in Heterogeneous Networks

Next generation wireless networks face the challenge of increasing energy consumption while satisfying the unprecedented increase in the data rate demand. To address this problem, we propose a utility-based energy-efficient resource allocation algorithm for the downlink transmissions in heterogeneous networks (HetNets). We consider the fractional frequency reuse (FFR) method in order to mitigate the intra- and inter-cell interference. The proposed algorithm divides the resource allocation problem into frequency and power assignment problems and sequentially solves them. The proposed power control algorithm uses the gradient ascent method to control the transmit power of macrocell base stations (MeNBs) as most of the power in the network is consumed there. We present the optimality conditions of the resource allocation problem and the convergence of the proposed algorithm. In order to mitigate the inter-cell interference further, we study the interference pricing mechanisms and obtain an upper bound to the maximum energy efficiency problem including the inter-cell interference contributions. The performance of the proposed algorithm is studied in a Long Term Evolution (LTE) system. Our simulation results demonstrate that the proposed algorithm provides substantial improvements in the energy efficiency and throughput of the network. It is also shown that interference pricing provides only marginal improvements over the proposed algorithm.

Solution of the determinantal assignment problem using the Grassmann matrices

ABSTRACTThe paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation where is an n −dimensional vector space over which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of , and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector are given in terms of the rank properties of the Grassmann matrix, of the vector , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if where ; the solution space for a decomposable , is the space . This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge–Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge–Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist.

통합 필터 변별도와 그래프 컬러링을 이용한 전술통신망 백본 무선 링크의 주파수 지정 방법

The tactical communications network has to be deployed rapidly at military operation area and support the communications between the military command systems and the weapon systems. For that, the frequency assignment is required for backbone wireless links of tactical communications network without frequency interferences. In this paper, we propose a frequency assignment method using net filter discrimination (NFD) and graph coloring to avoid frequency interferences. The proposed method presents frequency assignment problem of tactical communications network as vertex graph coloring problem of a weighted graph. And it makes frequency assignment sequences and assigns center frequencies to communication links according to the priority of communication links and graph coloring. The evaluation shows that this method can assign center frequencies to backbone communication links without frequency interferences. It also shows that the method can improve the frequency utilization in comparison with HTZ-warfare that is currently used by Korean Army.