Graph coloring: a novel heuristic based on trailing path—properties, perspective and applications in structured networks
- 4 Downloads
Graph coloring is a manifestation of graph partitioning, wherein a graph is partitioned based on the adjacency of its elements. The fact that there is no general efficient solution to this problem that may work unequivocally for all graphs opens up the realistic scope for combinatorial optimization algorithms to be invoked. The algorithmic complexity of graph coloring is non-deterministic in polynomial time and hard. To the best of our knowledge, there is no algorithm as yet that procures an exact solution of the chromatic number comprehensively for any and all graphs within the polynomial (P) time domain. Here, we present a novel heuristic, namely the ‘trailing path’, which returns an approximate solution of the chromatic number within P time, and with a better accuracy than most existing algorithms. The ‘trailing path’ algorithm is effectively a subtle combination of the search patterns of two existing heuristics (DSATUR and largest first) and operates along a trailing path of consecutively connected nodes (and thereby effectively maps to the problem of finding spanning tree(s) of the graph) during the entire course of coloring, where essentially lies both the novelty and the apt of the current approach. The study also suggests that the judicious implementation of randomness is one of the keys toward rendering an improved accuracy in such combinatorial optimization algorithms. Apart from the algorithmic attributes, essential properties of graph partitioning in random and different structured networks have also been surveyed, followed by a comparative study. The study reveals the remarkable stability and absorptive property of chromatic number across a wide array of graphs. Finally, a case study is presented to demonstrate the potential use of graph coloring in protein design—yet another hard problem in structural and evolutionary biology.
KeywordsChromatic number Graph partitioning NP to P Motif identifier Protein design
The work was supported by the Department of Science and Technology—Science and Engineering Research Board (DST-SERB research Grant PDF/2015/001079). We take the opportunity to thank Mr. Arnab Kar (Department of IT, IIIT Alahabad) for his brief participation during the revision.
SB conceived the problem. AB and SB designed the algorithm. AB wrote the initial MATLAB code which was improved at different stages by both AB and SB. For the analysis, AB provided small scripts which were executed by SB to carry out the computational experiments. AB and SB analyzed the results. SB wrote the paper with help from AB. AKD participated in the comparison with other heuristics and provided crucial notes at different portions of the manuscript. All authors read and approved the final manuscript.
Compliance with ethical standards
Conflict of interest
None of the authors have any competing interests in the manuscript.
- Albertson MO, Cranston DW, Fox J (2010) Crossings, colorings, and cliques. ArXiv10063783 MathGoogle Scholar
- Andreev K, Räcke H (2004) Balanced graph partitioning. In: Proceedings of the sixteenth annual ACM symposium on parallelism in algorithms and architectures. ACM, New York, pp 120–124Google Scholar
- Basu S, Bhattacharyya D, Banerjee R (2014) Applications of complementarity plot in error detection and structure validation of proteins. Indian J Biochem Biophys 51:188–200Google Scholar
- Crick FHC, IUCr (1953) The packing of -helices: simple coiled-coils. In: Acta crystallogr. http://scripts.iucr.org/cgi-bin/paper?S0365110X53001964. Accessed 30 Nov 2016
- Garey MR, Johnson DS, Stockmeyer L (1974) Some simplified NP-complete problems. In: Proceedings of the sixth annual ACM symposium on theory of computing. ACM, New York, pp 47–63Google Scholar
- Hansen J, Kubale M, Kuszner Ł, Nadolski A (2004) Distributed largest-first algorithm for graph coloring. In: Euro-Par 2004 parallel processing. Springer, Berlin, Heidelberg, pp 804–811Google Scholar
- Kosowski A, Manuszewski K (2004) Classical coloring of graphs. In: Graph colorings, pp 2–19Google Scholar
- Marx D (2003) Graph colouring problems and their applications in schedulingGoogle Scholar
- McIlvaine TC (1921) A buffer solution for colorimetric comparison. J Biol Chem 49:183–186Google Scholar
- RJLipton + KWRegan (2015) A big result on graph isomorphism. In: Gödels Lost Lett. PNP. https://rjlipton.wordpress.com/2015/11/04/a-big-result-on-graph-isomorphism/. Accessed 30 Nov 2016
- Zarrazola E, Gomez D, Montero J et al (2011) Network clustering by graph coloring: an application to astronomical images. In: 2011 11th international conference on intelligent systems design and applications (ISDA), pp 796–801Google Scholar