Frequency Assignment Problem Research Articles

Static beam placement and frequency plan algorithms for LEO constellations

SummaryLEO satellites are growing rapidly in number, but also in configurability of the payload, which presents new challenges of how to allocate the satellite's resources. We present an algorithm to solve two subproblems of the resource allocation problem for multi‐beam satellites in large LEO constellations. Our approach produces a static beam placement and a static frequency plan, taking into account the user demands and interference conditions, respectively. We provide an easy to implement and fast approximation for the beam placement problem, which is converted into an Edge Clique Cover problem using graph theory. The frequency assignment problem is translated into a constraint satisfaction problem, which is later solved with a first‐fit heuristic. We apply our algorithm to a scenario based on SpaceX's Starlink constellation, and when compared to two different baselines, we show that our method requires 40% fewer beams while being able to provide service to 2% more users.

A receptance-based method for frequency assignment via coupling of subsystems

This paper presents a theoretical study of the frequency assignment problem of a coupled system via structural modification of one of its subsystems. It deals with the issue in which the available modifications are not simple; for example, they are not point masses, grounded springs, or spring-mass oscillators. The proposed technique is derived based on receptance coupling technique and formulated as an optimization problem. Only a few receptances at the connection ends of each subsystem are required in the structural modification process. The applicability of the technique is demonstrated on a simulated rotor system. The results show that both bending natural frequencies and torsional natural frequencies can be assigned using a modifiable joint, either separately or simultaneously. In addition, an extension is made on a previously proposed torsional receptance measurement technique to estimate the rotational receptance in bending. Numerical simulations suggest that the extended technique is able to produce accurate estimations and thus is appropriate for this frequency assignment problem of concern.

An efficient hybrid multi-objective memetic algorithm for the frequency assignment problem

This paper investigates hybridization of multi-objective memetic algorithm and artificial immune system (AIS) for the Frequency Assignment Problem (FAP) in cellular mobile networks. The considered objectives to minimize are the total interference, the maximal interference, and the number of used frequencies. The proposed approach integrates FAP-specific local search into the evolutionary process to overcome the shortcoming of the multi-objective genetic algorithm, as well as clonal selection and receptor editing, which aims to improve the algorithm exploration and exploitation abilities. Based on the hypervolume metric, the proposed hybrid multi-objective algorithm produces high quality solutions as proved by the tests performed over COST259 instances and corroborated by the comparisons with the most frequently referred algorithms in the related literature. Furthermore, the effect and the behaviour of the main parameters of our algorithm and the interaction between them are analysed using the Design of Experiment (DOE).

An effective multi-objective hybrid immune algorithm for the frequency assignment problem

This paper presents a novel multi-objective hybrid metaheuristic that combines an artificial immune system (AIS) with genetic operators to address the multi-objective Frequency Assignment Problem (FAP) in cellular mobile networks, seeking to minimize three objectives simultaneously: Total interference, Maximal interference, and the number of used frequencies. The proposed approach inherits its evolutionary cycle from a multi-objective immune network supplemented by a FAP-specific local search as well as clonal selection for a better intensification. Moreover, a genetic mutation is applied at each generation to enhance the exploration of the research space. Computational experiments performed over COST259 instances and based on the hypervolume metric, show the efficiency of the newly proposed hybrid algorithm, and corroborated by the comparisons we did with the most frequently referred algorithms in the related literature. Furthermore, the effect of the main parameters and the interaction between them is analyzed using statistical tools.

Anti-k-labeling of graphs

It is well known that the labeling problems of graphs arise in many (but not limited to) networking and telecommunication contexts. In this paper we introduce the anti-k-labeling problem of graphs which we seek to minimize the similarity (or distance) of neighboring nodes. For example, in the fundamental frequency assignment problem in wireless networks where each node is assigned a frequency, it is usually desirable to limit or minimize the frequency gap between neighboring nodes so as to limit interference.Let k ≥ 1 be an integer and ψ is a labeling function (anti-k-labeling) from V(G) to {1,2,…,k} for a graph G. A no-hole anti-k-labeling is an anti-k-labeling using all labels between 1 and k. We define wψ(e)=|ψ(u)−ψ(v)| for an edge e=uv and wψ(G)=min{wψ(e):e∈E(G)} for an anti-k-labeling ψ of the graph G. The anti-k-labeling number of a graph G, λk(G), is max {wψ(G): ψ}. In this paper, we first show that λk(G)=⌊k−1χ−1⌋, and the problem that determines the anti-k-labeling number of graphs is NP-hard. We mainly obtain the lower bounds on no-hole anti-n-labeling number for trees, grids and n-cubes.

Three Local Search Meta-Heuristics for the Minimum Interference Frequency Assignment Problem (MI-FAP) in Cellular Networks

The minimum interference frequency assignment problem (MI-FAP) plays an important role in cellular networks. MI-FAP is the problem of finding an assignment of a small number of frequencies to a large number of transceivers (TRXs) that minimizes the interferences level. The MI-FAP is known to be NP-Hard, thus it cannot be solved in polynomial time. To remedy this, researchers usually use meta-heuristic techniques to find an approximate solution in reasonable time. Here, the authors propose three meta-heuristics for the MI-FAP: a variable neighborhood search (VNS) and a stochastic local search (SLS) that are combined to obtain a third and a new one, which is called VNS-SLS. The SLS method is incorporated into the VNS process as a subroutine in order to enhance the solution quality. All three proposed methods are evaluated on some well-known datasets to measure their performance. The empirical experiments show that the proposed method VNS-SLS succeeds in finding good results compared to both VNS and SLS confirming a good balance between intensification and diversification.

On the radio number for corona of paths and cycles

Radio k-coloring of graphs is one of the variations of frequency assignment problem. For a simple connected graph G and a positive integer k⩽diam(G), a radio k-coloring is an assignment f of positive integers (colors) to the vertices of G such that for every pair of distinct vertices u and v of G, the difference between their colors is at least 1+k−d(u,v). The maximum color assigned by f is called its span, denoted by rck(f). The radio k-chromatic number rck(G) of G is min{rck(f):fis a radiok-coloring ofG}. If d is the diameter of G, then a radio d-coloring is referred as a radio coloring and the radio d-chromatic number as the radio number, denoted by rn(G), of G. The corona G⊙H of two graphs G and H is the graph obtained by taking one copy of G and |V(G)| copies of H, and joining each and every vertex of the ith copy of H with the ith vertex of G by an edge. In this paper, for path Pn and cycle Cm, m≥5, we determine rn(Pn⊙Cm) when n is even, and give an upper bound for the same when n is odd. Also, for m≥4, we determine the radio number of Pn⊙Pm when n is even, and give both upper and lower bounds for rn(Pn⊙Pm) when n is odd.

Conflict-Free Coloring of Intersection Graphs of Geometric Objects

In 2002, Even et al. introduced and studied the notion of conflict-free colorings of geometrically defined hypergraphs. They motivated it by frequency assignment problems in cellular networks. This notion has been extensively studied since then. A conflict-free coloring of a graph is a coloring of its vertices such that the neighborhood (pointed or closed) of each vertex contains a vertex whose color differs from the colors of all other vertices in that neighborhood. In this paper we study conflict-free colorings of intersection graphs of geometric objects. We show that any intersection graph of n pseudo-discs in the plane admits a conflict-free coloring with $$O(\log n)$$ colors, with respect to both closed and pointed neighborhoods. We also show that the latter bound is asymptotically sharp. Using our methods, we obtain the following strengthening of the two main results of Even et al.: Any family $$\mathcal {F}$$ of n discs in the plane can be colored with $$O(\log n)$$ colors in such a way that for any disc B in the plane, not necessarily from $$\mathcal {F}$$ , the set of discs in $$\mathcal {F}$$ that intersect B contains a uniquely-colored element. In view of the original motivation to study such colorings, this strengthening suggests further applications to frequency assignment in wireless networks. Finally, we present bounds on the number of colors needed for conflict-free colorings of other classes of intersection graphs, including intersection graphs of axis-parallel rectangles and of $$\rho $$ -fat objects in the plane.

A Coral Reefs Optimization algorithm with substrate layer for robust Wi-Fi channel assignment

In this paper, we tackle a problem of frequency assignment in Wi-Fi networks with a novel evolutionary-type algorithm. In this version of the problem, we consider the interferences originated by the access points, and also by the clients and all the 11 available channels in the 2.4 GHz Wi-Fi frequency band. The proposed evolutionary-type algorithm is the Coral Reefs Optimization approach with substrate layer (CRO-SL). It is a recently proposed algorithm, which simulates the processes which occur in real coral reefs, including the reproduction and fight for the space of living corals. This version of the algorithm includes a layer of “substrates” which allows using different search patterns jointly in the algorithm. This way, the CRO-SL is able to apply search patterns such as harmony search, differential evolution, Gaussian-based mutations and other traditional and novel search procedures, including local search algorithms, within a single population of solutions. We show the good performance of the proposed approach in a real case study of Wi-Fi frequency assignment, in the Polytechnic School building of the Universidad de Alcala (Spain), where different realistic scenarios of the problem have been simulated and successfully solved with the CRO-SL algorithm.

A Novel Multi-Objective Immune Memetic Algorithm for the Frequency Assignment Problem

This paper presents a multi-objective immune memetic algorithm to the challenge of solving the Frequency Assignment Problem (FAP) in cellular networks seeking the minimization of the network’s total interference, the maximum interference and the number of used frequencies. The originality of the proposed approach lies in integrating a FAP-specific local search into its evolutionary process instead of crossover and mutation, as well as a guided diversification strategy for better performances. Moreover, the algorithm is supplemented with a clonal selection, inherited from Artificial Immune Systems (AIS), which aims to improve the algorithm exploration and exploitation abilities. Computational experiments performed over COST259 instances show the efficiency of the newly proposed evolutionary multi-objective algorithm and corroborated by the comparisons we did with the most frequently referred algorithm in the related literature. Furthermore, the effect of the main parameters and the interaction between them is analyzed using statistical tools.

Breakout variable neighbourhood search for the minimum interference frequency assignment problem

PurposeThis paper aims to describe two enhancements of the variable neighbourhood search (VNS) algorithm to solve efficiently the minimum interference frequency assignment problem (MI-FAP) which is a major issue in the radio networks, as well as a well-known NP-hard combinatorial optimisation problem. The challenge is to assign a frequency to each transceiver of the network with limited or no interferences at all. Indeed, considering that the number of radio networks users is ever increasing and that the radio spectrum is a scarce and expensive resource, the latter should be carefully managed to avoid any interference.Design/methodology/approachThe authors suggest two new enhanced VNS variants for MI-FAP, namely, the iterated VNS (It-VNS) and the breakout VNS (BVNS). These two algorithms were designed based on the hybridising and the collaboration approaches that have emerged as two powerful means to solve hard combinatorial optimisation problems. Therefore, these two methods draw their strength from other meta-heuristics. In addition, the authors introduced a new mechanism of perturbation to enhance the performance of VNS. An extensive experiment was conducted to evaluate the performance of the proposed methods on some well-known MI-FAP datasets. Moreover, they carried out a comparative study with other metaheuristics and achieved the Friedman’s non-parametric statistical test to check the actual effect of the proposed enhancements.FindingsThe experiments showed that the two enhanced methods (It-VNS) and (BVNS) achieved better results than the VNS method. The comparative study with other meta-heuristics showed that the results are competitive and very encouraging. The Friedman’s non-parametric statistical test reveals clearly that the results of the three methods (It-VNS, BVNS and VNS) are significantly different. The authors therefore carried out the Nemenyi’s post hoc test which allowed us to identify those differences. The impact of the operated change on both the It-VNS and BVNS was thus confirmed. The proposed BVNS is competitive and able to produce good results as compared with both It-VNS and VNS for MI-FAP.Research limitations/implicationsApproached methods and particularly newly designed ones may have some drawbacks that weaken the results, in particular when dealing with extensive data. These limitations should therefore be eliminated through an appropriate approach with a view to design appropriate methods in the case of large-scale data.Practical implicationsThe authors designed and implemented two new variants of the VNS algorithm before carrying out an exhaustive experimental study. The findings highlighted the potential opportunities of these two enhanced methods which could be adapted and applied to other combinatorial optimisation problems, real world applications or academic problems.Originality/valueThis paper aims at enhancing the VNS algorithm through two new approaches, namely, the It-VNS and the BVNS. These two methods were applied to the MI-FAP which is a crucial problem arising in a radio network. The numerical results are interesting and demonstrate the benefits of the proposed approaches in particular BVNS for MI-FAP.

Parasitic Optimisation Algorithm: a Lesson from Striga Asiatica (L) Kuntze Weed

In this paper, an attempt to formulate a new meta-heuristic, called Parasitic Optimisation algorithm, is carried out for solving network optimisation problems. The algorithm is based on how the natural striga asiatica weed infects the host plant and its survival tactics. The proposed algorithm is validated using two examples of looped water networks, 18 generalised traveling salesman, 35 capacitated vehicle routing and 35 dynamic frequency assignment problem instances. The results are compared to those existing in the literature. The results show that the new algorithm seems to perform better as compared to those existing in literature in-terms of execution time and convergence to the optimum solution. Future research should focus on parameter studies of the algorithm. Parastic Optimisation Striga Asiatica Algorithm

A Neighborhood Search-Based Heuristic for the Fixed Spectrum Frequency Assignment Problem

This article proposes a heuristic for the fixed spectrum frequency assignment (FS-FA) problem of telecommunications networks. A network composes of many connections, and each connection needs a frequency from the spectrum. The assignment of frequencies to the transmitters should satisfy a set of constraints. The constraints specify the separation which is necessary between frequencies of different transmitters. Violation of constraints creates interference. The goal of the FS-FA problem is to find an assignment of frequencies for the transmitters, which has minimum interference. The proposed heuristic has two main components: a local search heuristic and a compound move. The local search heuristic employs one-change moves (i.e., a move that changes the frequency of one transmitter at a time). It also employs a lookup table that classifies all possible one-change moves as positive or negative. The local search heuristic chooses positive/negative moves until it traps in a locally minimal solution. The compound-move operation shifts the local search to a new location in the search space. We can repeatedly apply the local search and compound move for many iterations. The proposed heuristic has been evaluated on the same benchmarks as used by others in the recently published literature. We have compared our algorithm with two existing tabu-search-based algorithms: dynamic-list-based tabu search (DTS) (Montemanni et al. in IEEE Trans Veh Technol 52(4):891–901, 2003. https://doi.org/10.1109/TVT.2003.810976 ) and heuristic manipulation technique-based TS (Montemanni and Smith in Comput Oper Res 37(3):543–551, 2010. https://doi.org/10.1016/j.cor.2008.08.006 ) (HMT). The solution quality of the proposed algorithm is found to be better than or equal to the HMT and DTS in 88% and 79% of test problems, respectively.

Notes on complexity of packing coloring

A packing k-coloring for some integer k of a graph G=(V,E) is a mapping φ:V→{1,…,k} such that any two vertices u,v of color φ(u)=φ(v) are in distance at least φ(u)+1. This concept is motivated by frequency assignment problems. The packing chromatic number of G is the smallest k such that there exists a packing k-coloring of G.Fiala and Golovach showed that determining the packing chromatic number for chordal graphs is NP-complete for diameter exactly 5. While the problem is easy to solve for diameter 2, we show NP-completeness for any diameter at least 3. Our reduction also shows that the packing chromatic number is hard to approximate within n1/2−ε for any ε>0.In addition, we design an FPT algorithm for interval graphs of bounded diameter. This leads us to exploring the problem of finding a partial coloring that maximizes the number of colored vertices.

Natural frequency assignment for mass-chain systems with inerters

Abstract This paper studies the problem of natural frequency assignment for mass-chain systems with inerters. This is the problem to determine whether an arbitrary set of positive numbers may be assigned as the natural frequencies of a chain of n masses in which each element has fixed mass and is connected to its neighbour by a parallel combination of a spring and inerter. It is proved that mass-chain systems with inerters may have multiple natural frequencies, which is different from conventional mass-chain systems (without inerters) whose natural frequencies are always simple. It is shown that arbitrary assignment of natural frequencies including multiplicities is not possible with the choice of n inerters and n springs. In particular, it is shown that an eigenvalue of multiplicity m may occur only if n ⩾ 2 m - 1 . However, it is proved that n - 1 inerters and n springs are necessary and sufficient to freely assign an arbitrary set of distinct positive numbers as the natural frequencies of an n-degree-of-freedom mass-chain system.

Multi-Objectivising Combinatorial Optimisation Problems by Means of Elementary Landscape Decompositions.

In the last decade, many works in combinatorial optimisation have shown that, due to the advances in multi-objective optimisation, the algorithms from this field could be used for solving single-objective problems as well. In this sense, a number of papers have proposed multi-objectivising single-objective problems in order to use multi-objective algorithms in their optimisation. In this article, we follow up this idea by presenting a methodology for multi-objectivising combinatorial optimisation problems based on elementary landscape decompositions of their objective function. Under this framework, each of the elementary landscapes obtained from the decomposition is considered as an independent objective function to optimise. In order to illustrate this general methodology, we consider four problems from different domains: the quadratic assignment problem and the linear ordering problem (permutation domain), the 0-1 unconstrained quadratic optimisation problem (binary domain), and the frequency assignment problem (integer domain). We implemented two widely known multi-objective algorithms, NSGA-II and SPEA2, and compared their performance with that of a single-objective GA. The experiments conducted on a large benchmark of instances of the four problems show that the multi-objective algorithms clearly outperform the single-objective approaches. Furthermore, a discussion on the results suggests that the multi-objective space generated by this decomposition enhances the exploration ability, thus permitting NSGA-II and SPEA2 to obtain better results in the majority of the tested instances.

Perturbation-minimising frequency assignment to address short term demand fluctuation in cellular network

In cellular network short term demand fluctuation is a very common phenomenon. The demand of any cell may increase or decrease slightly or the system may expand by adding additional cells or the system may shrink if the demands of certain number of cells become zero. In this paper, the perturbation-minimising frequency assignment problem (PMFAP) is considered to address the short term fluctuation in demand vector. PMFAP is a frequency assignment problem in which newly generated demands are satisfied with minimum changes in the already existing frequency assignment keeping all the interference constraints. In this paper, an efficient heuristic algorithm for PMFAP is presented. The efficiency of this algorithm is compared with the existing results from literature. With a slight modification to the proposed algorithm, it can solve the well-known frequency assignment problem (FAP) and its performance is also compared with the existing results using the standard benchmark data sets for FAP.

Perturbation-minimising frequency assignment to address short term demand fluctuation in cellular network

In cellular network short term demand fluctuation is a very common phenomenon. The demand of any cell may increase or decrease slightly or the system may expand by adding additional cells or the system may shrink if the demands of certain number of cells become zero. In this paper, the perturbation-minimising frequency assignment problem (PMFAP) is considered to address the short term fluctuation in demand vector. PMFAP is a frequency assignment problem in which newly generated demands are satisfied with minimum changes in the already existing frequency assignment keeping all the interference constraints. In this paper, an efficient heuristic algorithm for PMFAP is presented. The efficiency of this algorithm is compared with the existing results from literature. With a slight modification to the proposed algorithm, it can solve the well-known frequency assignment problem (FAP) and its performance is also compared with the existing results using the standard benchmark data sets for FAP.

Nash-Pareto Genetic Algorithm for the Frequency Assignment Problem

This paper presents a hybrid multi-objective genetic algorithm which combines the main notion of game theory Nash equilibrium with Pareto-optimality to solve the multi-objective Frequency Assignment Problem (FAP) in mobile networks. The game is coupled with genetic algorithm to accelerate convergence and produce Nash equilibrium and Pareto non-dominated solutions simultaneously. The proposed hybrid approach produces high quality solutions as proved by several performed tests and corroborated by the comparison with the most referred multi-objective optimization algorithms such as NSGA-II and SPEA2 on well-known Philadelphia and COST259 FAP instances. Furthermore, the effect of some parameters is discussed.

<i>L</i>(2, 1)-labeling of the Cartesian and strong product of two directed cycles

The frequency assignment problem (FAP) is the assignment of frequencies to television and radio transmitters subject to restrictions imposed by the distance between transmitters. One of the graph theoretical models of FAP which is well elaborated is the concept of distance constrained labeling of graphs. Let $G = (V, E)$ be a graph. For two vertices $u$ and $v$ of $G$, we denote $d(u, v)$ the distance between $u$ and $v$. An $L(2, 1)$-labeling for $G$ is a function $f: V → \{0, 1, ···\}$ such that $|f(u)-f(v)| ≥ 1$ if $d(u, v) = 2$ and $|f(u)-f(v)| ≥ 2$ if $d(u, v) = 1$. The span of $f$ is the difference between the largest and the smallest number of $f(V)$. The $λ$-number for $G$, denoted by $λ(G)$, is the minimum span over all $L(2, 1)$-labelings of $G$. In this paper, we study the $λ$-number of the Cartesian and strong product of two directed cycles. We show that for $m, n ≥ 4$ the $λ$-number of $\overrightarrow{C_m} \Box \overrightarrow{C_n}$ is between 4 and 5. We also establish the $λ$-number of $\overrightarrow{{{C}_{m}}}\boxtimes \overrightarrow{{{C}_{n}}}$ for $m ≤ 10$ and prove that the $λ$-number of the strong product of cycles $\overrightarrow{{{C}_{m}}}\boxtimes \overrightarrow{{{C}_{n}}}$ is between 6 and 8 for $m, n ≥ 48$.