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Graph Coloring: a Novel Heuristic Based on Trailing Path—Properties, Perspective and Applications in Structured Networks Graph coloring: a novel heuristic based on trailing path—properties, perspective and applications in structured networks Abhirup Bandyopadhyay, Amit kumar Dhar & Sankar Basu Soft Computing A Fusion of Foundations, Methodologies and Applications ISSN 1432-7643 Volume 24 Number 1 Soft Comput (2020) 24:603-625 DOI 10.1007/s00500-019-04278-8 1 23 Your article is protected by copyright and all rights are held exclusively by Springer- Verlag GmbH Germany, part of Springer Nature. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Soft Computing (2020) 24:603–625 https://doi.org/10.1007/s00500-019-04278-8 (0123456789().,-volV)(0123456789().,- volV) METHODOLOGIES AND APPLICATION Graph coloring: a novel heuristic based on trailing path—properties, perspective and applications in structured networks 1 2,3 4,5,6 Abhirup Bandyopadhyay • Amit kumar Dhar • Sankar Basu Published online: 14 August 2019 Ó Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract 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 prop- erties of graph partitioning in random and different structured networks have also been surveyed, followed by a com- parative 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. Keywords Chromatic number Á Graph partitioning Á NP to P Á Motif identifier Á Protein design 1 Introduction In graph theory, graph coloring (Jensen and Toft 2011)isa Communicated by V. Loia. special case of graph labeling (Dı´az et al. 2002). It is an assignment of labels (Gallian 2015) traditionally known as Abhirup Bandyopadhyay and Sankar Basu have contributed equally to this work. ‘colors’ to edges and/or vertices of a graph subject to certain constraints. In trivial formalism, it is a way of Electronic supplementary material The online version of this coloring the vertices (nodes) of an undirected graph such article (https://doi.org/10.1007/s00500-019-04278-8) con- that no two adjacent vertices could be assigned the same tains supplementary material, which is available to autho- rized users. 3 Present Address: Department of EECS, IIT Bhilai, & Sankar Basu Raypur 492015, India [email protected] 4 Department of Physics and Astronomy, Clemson University, 1 Department of Mathematics, National Institute of Clemson, SC, USA Technology, Durgapur, Mahatma Gandhi Avenue, Durgapur, 5 Present Address: 3BIO, ULB, 1050 Brussels, Belgium West Bengal 713209, India 6 Department of Microbiology, Asutosh College, 2 Department of IT, IIIT Alahabad, Jhalwa, Alahabad 211012, Kolkata 700026, India India 123 Author's personal copy 604 A. Bandyopadhyay et al. color. This is called vertex coloring (MacDougall et al. Lova´sz conjecture bounding the chromatic number of 2002). Similarly, an edge coloring (Wallis et al. 2000) unions of complete graphs that have exactly one vertex in assigns a color to each edge so that no two adjacent edges common to each pair, and the Albertson conjecture share the same color, and a face coloring of a planar graph (Albertson et al. 2010) that among k-chromatic graphs, the (Sanders and Zhao 2001) assigns colors to each face or complete graphs are the ones with the smallest crossing region so that no two faces which share a common number. boundary share the same color. Given all this, vertex col- The first results about graph coloring deal almost oring remains the first chapter of the subject, and other exclusively with planar graphs in the form of the coloring coloring problems are transformable into a vertex version. of maps (Stiebitz and Sˇkrekovski 2006). While working on For example, an edge coloring of a graph is actually a the map coloring problem of the counties of England, vertex coloring of the corresponding line graph, and a face Francis Guthrie postulated the four color conjecture, noting coloring of a planar graph is a vertex coloring of its dual that four colors were sufficient to color a map so that no graph. However, non-vertex coloring problems are often regions sharing a common border receives the same color. stated and studied independently. That is partly because of In 1879, Alfred Kempe published a paper (Kempe 1879) perspective, and partly because some problems when nat- that claimed to establish the result which was controversial, urally extended could be best studied in the form of edges. followed by much debate. In fact it took close to a century The convention of using colors was originated from until the four-color theorem was finally proved in 1976 by coloring the countries of a geographical map, where each Kenneth Appel and Wolfgang Haken (Appel and Haken face is literally colored. This particular problem was for- 1977). The proof was the first major computer-aided proof malized as coloring the faces of a planar graph. By in this problem which went back to the ideas of Heawood implementing ‘planar duality’, a characteristic feature of and Kempe while largely disregarding the intervening planar graphs, the problem reduces to coloring of their developments. From that time onwards, active research is vertices. As a generalization the face coloring problem ongoing on the algorithmic attributes of graph coloring. could be viewed as vertex coloring problem of its dual The chromatic number problem falls in the list of Karp’s 21 graph. For the sake of simplicity and computational effi- NP-complete problems (Karp 1972) and remains compu- ciency, first few positive or nonnegative integers are used tationally NP-hard (Garey et al. 1974). That is to say that it as the ‘colors’ (Zhang 2015) without the loss of generality, is NP-complete to decide whether a given graph admits a k- so that one can use any finite set of colors as the ‘color set’. coloring for any given k except for the trivial cases Graph coloring could be viewed as the problem of k [ {0,1,2}. In other words, the 3-coloring problem assigning colors to a graph subject to number of con- remains NP-complete even on 4-regular planar graphs straints. Different constraints could range from constraints (Dailey 1980), and the approximation algorithm (Hallo´rs- on a subgraph to those on the full graph or even on the son 1993), the most established one in the field, computes a color itself. The face coloring problem even attained pop- coloring of graph size n at most within a factor of O(n(log ularity among common people in the form of the popular n)-3(log (log (n)))2) of the chromatic number. number puzzle Sudoku, the traveling salesman problem The relatively recent concept of chromatic polynomial and the Chinese postman problem. One of the major (Dong et al. 2005) has provided another alternative applications of graph coloring is the register allocation in approach toward solving the graph coloring problem, compilers. The range of applications grows even further serving important fundamental structures in algebraic ranging from coding theory to X-ray Crystallography graph theory. However, nowadays, the most celebrated (Blum et al. 1987), from radar and astronomy (Zarrazola conjecture is perhaps the ‘strong perfect graph conjecture’, et al. 2011) to circuit design and communication networks. which was first brought about by Claude Berge, originally Day-to-day real-life problems like guarding an art gallery, motivated by an information-theoretic concept called the physical layout segmentation, round robin sports schedul- ‘zero error capacity’ (Lovasz 2006) of a graph introduced ing, aircraft scheduling (Marx 2003), etc., should poten- by Claude E. Shannon. tially be benefited by an elegant algorithmic solution of the The recent literature of combinatorial optimization problem. Graph coloring is still a very active area of problems in general consists of a wide variety of related yet research with a bunch of unsolved problems, e.g., the distinct approaches ranging from adaptive and evolutionary chromatic number of the plane is unknown where two algorithms [including the implementation of multi-variant points are adjacent if they have unit distance.
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