Frequency assignment: Theory and applications

In this paper we introduce the minimum-order approach to frequency assignment and present a theory which relates this approach to the traditional one. This new approach is potentially more desirable than the traditional one. We model assignment problems as both frequency-distance constrained and frequency constrained optimization problems. The frequency constrained approach should be avoided if distance separation is employed to mitigate interference. A restricted class of graphs, called disk graphs, plays a central role in frequency-distance constrained problems. We introduce two generalizations of chromatic number and show that many frequency assignment problems are equivalent to generalized graph coloring problems. Using these equivalences and recent results concerning the complexity of graph coloring, we classify many frequency assignment problems according to the "execution time efficiency" of algorithms that may be devised for their solution. We discuss applications to important real world problems and identify areas for further work.

[1]  R. L. Brooks.  On Colouring the Nodes of a Network , 1941 .

[2]  P. Armttage,et al.  An overlap problem arising in particle counting. , 1949 .

[3]  M. S. Bartlett,et al.  The expected number of clumps when convex laminae are placed at random and with random orientation on a plane area , 1954, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  E. A. Nordhaus,et al.  On Complementary Graphs , 1956 .

[5]  E. N. Gilbert,et al.  Random Plane Networks , 1961 .

[6]  E. Gilbert.  The probability of covering a sphere with N circular caps , 1965 .

[7]  J. E. L. Peck,et al.  Algorithm 286: Examination scheduling , 1966, CACM.

[8]  H. Wilf.  The Eigenvalues of a Graph and Its Chromatic Number , 1967 .

[9]  D. J. A. Welsh,et al.  An upper bound for the chromatic number of a graph and its application to timetabling problems , 1967, Comput. J..

[10]  G. Szekeres,et al.  An inequality for the chromatic number of a graph , 1968 .

[11]  David C. Wood,et al.  A technique for colouring a graph applicable to large scale timetabling problems , 1969, Computer/law journal.

[12]  P. Holgate.  Majorants of the Chromatic Number of a Random Graph , 1969 .

[13]  J. Bondy.  Bounds for the chromatic number of a graph , 1969 .

[14]  R. S. Wilkov,et al.  A practical approach to the chromatic partition problem , 1970 .

[15]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[16]  The coloring of graphs in a linear number of steps , 1971 .

[17]  Nicos Christofides,et al.  An Algorithm for the Chromatic Number of a Graph , 1971, Comput. J..

[18]  B. R. Myers,et al.  A lower bound on the chromatic number of a graph , 1971, Networks.

[19]  J. R. Brown.  Chromatic Scheduling and the Chromatic Number Problem , 1972 .

[20]  D. Matula,et al.  GRAPH COLORING ALGORITHMS , 1972 .

[21]  Fanica Gavril,et al.  Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph , 1972, SIAM J. Comput..

[22]  David W. Matula,et al.  Bounded color functions on graphs , 1972, Networks.

[23]  Amir Pnueli,et al.  Permutation Graphs and Transitive Graphs , 1972, JACM.

[24]  Richard M. Karp,et al.  Reducibility among combinatorial problems" in complexity of computer computations , 1972 .

[25]  J. Arthur Zoellner.  Frequency Assignment Games and Strategies , 1973 .

[26]  Bruce P. Graham,et al.  An Algorithm for Determining the Chromatic Number of a Graph , 1973, SIAM J. Comput..

[27]  Antonio L. Furtado,et al.  An Algorithm for Obtaining the Chromatic Number and an Optimal Coloring of a Graph , 1973, Information Processing Letters.

[28]  A. Tucker,et al.  Perfect Graphs and an Application to Optimizing Municipal Services , 1973 .

[29]  Chung C. Wang,et al.  An Algorithm for the Chromatic Number of a Graph , 1974, JACM.

[30]  David S. Johnson,et al.  Some simplified NP-complete problems , 1974, STOC '74.

[31]  Michael R. Williams.  Heuristic procedures (if they work—leave them alone) , 1974, Softw. Pract. Exp..

[32]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.

[33]  Paul Erdös,et al.  On the connection between chromatic number, maximal clique and minimal degree of a graph , 1974, Discret. Math..

[34]  Robert Frazier.  Compatibility and the Frequency Selection Problem , 1975, IEEE Transactions on Electromagnetic Compatibility.

[35]  J. Durkin,et al.  Computer prediction of field strength in the planning of radio systems , 1975, IEEE Transactions on Vehicular Technology.

[36]  A. Tucker,et al.  Coloring a Family of Circular Arcs , 1975 .

[37]  Eugene L. Lawler,et al.  A Note on the Complexity of the Chromatic Number Problem , 1976, Inf. Process. Lett..

[38]  David S. Johnson,et al.  The Complexity of Near-Optimal Graph Coloring , 1976, J. ACM.

[39]  John Mitchem,et al.  On Various Algorithms for Estimating the Chromatic Number of a Graph , 1976, Comput. J..

[40]  T. Sakaki,et al.  Algorithms for Finding in the Lump Both Bounds of Chromatic Number of a Graph , 1976, Comput. J..

[41]  Scott Cameron.  The Solution of the Graph-Coloring Problem as a Set-Covering Problem , 1977, IEEE Transactions on Electromagnetic Compatibility.

[42]  J. Zoeliner,et al.  A Breakthrough in Spectrum Conserving Frequency Assignment Technology , 1977, IEEE Transactions on Electromagnetic Compatibility.

[43]  Paul Major.  A Parameter-Sensitive Frequency-Assignment Method (PSFAM) , 1977, IEEE Transactions on Electromagnetic Compatibility.

[44]  C.L. Beall,et al.  An Automated Frequency Assignment System for Air Voice Communication Circuits in the Frequency Band from 225 MHz to 400 MHz , 1977, 1977 IEEE International Symposium on Electromagnetic Compatibility.

[45]  Walter A. Burkhard,et al.  Efficient algorithms for (3, 1) graphs , 1977, Inf. Sci..

[46]  F. Box,et al.  A heuristic technique for assigning frequencies to mobile radio nets , 1978, IEEE Transactions on Vehicular Technology.

[47]  Paul A. Catlin,et al.  Another bound on the chromatic number of a graph , 1978, Discret. Math..

[48]  A. K. Choudhury,et al.  A new method to find out the chromatic partition of a symmetric graph , 1978 .

[49]  Jim Lawrence.  Covering the vertex set of a graph with subgraphs of smaller degree , 1978, Discret. Math..

[50]  C. Papadimitriou,et al.  The Efficiency of Algorithms. , 1978 .

[51]  Paul A. Catlin.  A bound on the chromatic number of a graph , 1978, Discret. Math..

[52]  Lawrence Middlekamp.  UHF Taboos-History and Development , 1978, IEEE Transactions on Consumer Electronics.

[53]  Colin McDiarmid,et al.  Determining the Chromatic Number of a Graph , 1979, SIAM J. Comput..

[54]  S. Cameron,et al.  A Frequency Assignment Algorithm Based on a Minimum Residual Difficulty Heuristic , 1979, 1979 IEEE International Symposium on Electromagnetic Compatibility.

[55]  N. J. A. Sloane,et al.  New Bounds on the Number of Unit Spheres That Can Touch a Unit Sphere in n Dimensions , 1979, J. Comb. Theory, Ser. A.

[56]  Daniel Brélaz,et al.  New methods to color the vertices of a graph , 1979, CACM.

[57]  Gary L. Miller,et al.  The Complexity of Coloring Circular Arcs and Chords , 1980, SIAM J. Algebraic Discret. Methods.

[58]  Alan C. Tucker,et al.  An Efficient Test for Circular-Arc Graphs , 1980, SIAM J. Comput..

[59]  D. P. Bovet,et al.  An $O(n^2 )$ Algorithm for Coloring Proper Circular Arc Graphs , 1981 .