A learning approach to the bandwidth multicolouring problem

In this article, a generalisation of the vertex colouring problem known as bandwidth multicolouring problem (BMCP), in which a set of colours is assigned to each vertex such that the difference between the colours, assigned to each vertex and its neighbours, is by no means less than a predefined threshold, is considered. It is shown that the proposed method can be applied to solve the bandwidth colouring problem (BCP) as well. BMCP is known to be NP-hard in graph theory, and so a large number of approximation solutions, as well as exact algorithms, have been proposed to solve it. In this article, two learning automata-based approximation algorithms are proposed for estimating a near-optimal solution to the BMCP. We show, for the first proposed algorithm, that by choosing a proper learning rate, the algorithm finds the optimal solution with a probability close enough to unity. Moreover, we compute the worst-case time complexity of the first algorithm for finding a 1/(1–ϵ) optimal solution to the given prob...

[1]  Heiko Schröder,et al.  Evolutionary graph colouring , 2000, Inf. Process. Lett..

[2]  Steven Skiena,et al.  Coloring Graphs With a General Heuristic Search Engine , 2002 .

[3]  Tommy R. Jensen,et al.  Graph Coloring Problems: Jensen/Graph , 1994 .

[4]  D. de Werra,et al.  An introduction to timetabling , 1985 .

[5]  M. Thathachar,et al.  A Hierarchical System of Learning Automata , 1981, IEEE Transactions on Systems, Man, and Cybernetics.

[6]  Chirag M. Patel,et al.  An ant-based algorithm for coloring graphs , 2008, Discret. Appl. Math..

[7]  Steven Minton,et al.  Minimizing Conflicts: A Heuristic Repair Method for Constraint Satisfaction and Scheduling Problems , 1992, Artif. Intell..

[8]  Steven David Prestwich,et al.  Coloration Neighbourhood Search With Forward Checking , 2002, Annals of Mathematics and Artificial Intelligence.

[9]  Charles Fleurent,et al.  Genetic and hybrid algorithms for graph coloring , 1996, Ann. Oper. Res..

[10]  S. Lakshmivarahan,et al.  Absolute Expediency of Q-and S-Model Learning Algorithms , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[11]  Yi Zhu,et al.  Heuristic methods for graph coloring problems , 2005, SAC '05.

[12]  Vahid Lotfi,et al.  A graph coloring algorithm for large scale scheduling problems , 1986, Comput. Oper. Res..

[13]  Arie M. C. A. Koster,et al.  Transparent optical network design with sparse wavelength conversion , 2002 .

[14]  Martin E. Dyer,et al.  On Markov Chains for Randomly H-Coloring a Graph , 2001, J. Algorithms.

[15]  Mandayam A. L. Thathachar,et al.  Convergence of teams and hierarchies of learning automata in connectionist systems , 1995, IEEE Trans. Syst. Man Cybern..

[16]  Fred S. Roberts,et al.  T-colorings of graphs: recent results and open problems , 1991, Discret. Math..

[17]  B. R. Harita,et al.  Learning automata with changing number of actions , 1987, IEEE Transactions on Systems, Man, and Cybernetics.

[18]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[19]  Zaher Mahjoub,et al.  On a parallel genetic-tabu search based algorithm for solving the graph colouring problem , 2009, Eur. J. Oper. Res..

[20]  Kathryn A. Dowsland,et al.  An improved ant colony optimisation heuristic for graph colouring , 2008, Discret. Appl. Math..

[21]  Giuseppe F. Italiano,et al.  CHECKCOL: Improved local search for graph coloring , 2006, J. Discrete Algorithms.

[22]  Paolo Dell'Olmo,et al.  Coloring graphs by iterated local search traversing feasible and infeasible solutions , 2008, Discret. Appl. Math..

[23]  Michael A. Trick,et al.  A Column Generation Approach for Graph Coloring , 1996, INFORMS J. Comput..

[24]  Paolo Toth,et al.  An evolutionary approach for bandwidth multicoloring problems , 2008, Eur. J. Oper. Res..

[25]  Kumpati S. Narendra,et al.  On the Behavior of a Learning Automaton in a Changing Environment with Application to Telephone Traffic Routing , 1980, IEEE Transactions on Systems, Man, and Cybernetics.

[26]  Alain Hertz,et al.  A survey of local search methods for graph coloring , 2004, Comput. Oper. Res..

[27]  John Cocke,et al.  Register Allocation Via Coloring , 1981, Comput. Lang..

[28]  Pascal Brisset,et al.  Graph Coloring for Air Traffic Flow Management , 2004, Ann. Oper. Res..

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

[30]  Jan Karel Lenstra,et al.  On local search for the generalized graph coloring problem , 2003, Oper. Res. Lett..

[31]  Gilbert Laporte,et al.  Examination Timetabling: Algorithmic Strategies and Applications , 1994 .

[32]  Jürgen Peemöller,et al.  A correction to Brelaz's modification of Brown's coloring algorithm , 1983, CACM.

[33]  K. Narendra,et al.  Learning Algorithms for Two-Person Zero-Sum Stochastic Games with Incomplete Information: A Unified Approach , 1982 .

[34]  Steven David Prestwich,et al.  Generalised graph colouring by a hybrid of local search and constraint programming , 2008, Discret. Appl. Math..

[35]  Pedro M. Talaván,et al.  The graph coloring problem: A neuronal network approach , 2008, Eur. J. Oper. Res..

[36]  Kathryn A. Dowsland,et al.  A robust simulated annealing based examination timetabling system , 1998, Comput. Oper. Res..

[37]  Ben Paechter,et al.  Finding Feasible Timetables Using Group-Based Operators , 2007, IEEE Transactions on Evolutionary Computation.

[38]  Kumpati S. Narendra,et al.  Learning automata - an introduction , 1989 .

[39]  Jin-Kao Hao,et al.  Hybrid Evolutionary Algorithms for Graph Coloring , 1999, J. Comb. Optim..

[40]  Tommy R. Jensen,et al.  Graph Coloring Problems , 1994 .

[41]  Marek Kubale,et al.  A generalized implicit enumeration algorithm for graph coloring , 1985, CACM.

[42]  Yi Zhu,et al.  A Hybrid Method for the Graph Coloring and Related Problems , 2003 .

[43]  Javier Yáñez,et al.  Coloring fuzzy graphs , 2005 .

[44]  Nicolas Zufferey,et al.  A graph coloring heuristic using partial solutions and a reactive tabu scheme , 2008, Comput. Oper. Res..

[45]  John Cocke,et al.  A methodology for the real world , 1981 .

[46]  Jano I. van Hemert,et al.  Graph Coloring with Adaptive Evolutionary Algorithms , 1998, J. Heuristics.

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

[48]  Hishammuddin Asmuni,et al.  An investigation of fuzzy multiple heuristic orderings in the construction of university examination timetables , 2009, Comput. Oper. Res..

[49]  A. Gamst,et al.  Some lower bounds for a class of frequency assignment problems , 1986, IEEE Transactions on Vehicular Technology.

[50]  D. Werra,et al.  Some experiments with simulated annealing for coloring graphs , 1987 .

[51]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .