Abstract
In this paper a fast greedy sequential heuristic for the vertex colouring problem is presented. The suggested algorithm builds the same colouring of the graph as the well-known greedy sequential heuristic in which on every step the current vertex is coloured in the minimum possible colour. Our main contributions include introduction of a special matrix of forbidden colours and application of efficient bitwise operations on bit representations of the adjacency and forbidden colours matrices. Computational experiments show that in comparison with the classical greedy heuristic the average speedup of the developed approach is 2.6 times on DIMACS instances.
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References
Batsyn M, Goldengorin B, Maslov E, Pardalos PM (2014) Improvements to MCS algorithm for the maximum clique problem. J Comb Optim 27:397–416
Brelaz D (1979) New methods to color the vertices of a graph. Commun ACM 22:251–256
Briggs P, Cooper K, Torczon L (1992) Coloring register pairs. ACM Lett Program Lang Syst 1(1):3–13
Erciyes K (2013) Vertex coloring. In: distributed graph algorithms for computer networks, pp 107–134
Halldorsson MM (1997) Parallel and on-line graph coloring. J Algorithms 23:265–280
Johnson DS (1974) Worst case behavior of graph coloring algorithms. In: Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing. Utilitas Mathematica Publishing, pp 513–528
Karp Richard M (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations. Plenum, New York, pp 85–103
Kochenberger G, Glover F, Alidaee B, Wang H (2005) Clustering of microarray data via clique partitioning. J Comb Optim 10(1):77–92
Kosowski A, Manuszewski K (2004) Classical coloring of graphs. In: Kubale M (ed) Graph colorings. AMS Contemporary Mathematics, pp 1–20
Malaguti E, Monaci M, Toth P (2011) An exact approach for the vertex coloring problem. Discret Optim 8(2):174–190
Matula DM, Marble BG, Isaacson JD (1972) Graph coloring algorithms. In: Graph theory and computing. Academic Press, New York, pp 109–122
Meirong X, Yuzhen W (2014) T-coloring of graphs with application to frequency assignment in cellular mobile networks. In: Proceedings of the 33rd Chinese Control Conference (CCC), 2014
Méndez-Díaz I, Zabala P (2006) A Branch-and-cut algorithm for graph coloring. Discret Appl Math 154(5):826–847
MirHassani SA, Habibi F (2013) Solution approaches to the course timetabling problem. Artif Intell Rev 39:133–149
Odaira R, Nakaike T, Inagaki T, Komatsu H, Nakatani T (2010) Coloring-based coalescing for graph coloring register allocation. In: CGO ’10: Proceedings of the 8th annual IEEE/ACM international symposium on Code generation and optimization
Pardalos PM, Mavridou T, Xue J (1999) The graph coloring problem: a bibliographic survey. In: Du DZ, Pardalos PM (ed) Handbook of combinatorial optimization, pp 1077–1141
Porumbela DC, Hao JK, Kuntz P (2010) A search space “cartography” for guiding graph coloring heuristics. Comput Oper Res 37:769–778
Radin A (2000) Graph coloring heuristics from investigation of smallest hard to color graphs. MS Thesis, Rochester Institute of Technology
San Segundo P (2012) A new DSATUR-based algorithm for exact vertex coloring. Comput Oper Res 039(7):1724–1733
San Segundo P, Rodriguez-Losada D, Jimenez A (2011) An exact bit-parallel algorithm for the maximum clique problem. Comput Oper Res 038(2):571–581
Smith MD, Ramsey N, Holloway G (2004) A generalized algorithm for graph-coloring register allocation. In: PLDI ’04: Proceedings of the ACM SIGPLAN 2004 Conference on Programming language design and implementation
Wang H, Alidaee B, Kochenberger GA (2004) Evaluating a clique partitioning problem model for clustering high-dimensional data mining. In: Proceedings of the 10th Americas Conference on Information Systems (AMCIS 2004), New York, NY, 6–8 August 2004
Welsh DJA, Powell MB (1967) An upper bound for the chromatic number of a graph and its application to timetabling problems. Comput J 10(1):85–86
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The work is conducted at National Research University Higher School of Economics and supported by RSF Grant 14-41-00039.
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Komosko, L., Batsyn, M., Segundo, P.S. et al. A fast greedy sequential heuristic for the vertex colouring problem based on bitwise operations. J Comb Optim 31, 1665–1677 (2016). https://doi.org/10.1007/s10878-015-9862-1
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DOI: https://doi.org/10.1007/s10878-015-9862-1