Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-22T07:46:03.788Z Has data issue: false hasContentIssue false
  • English
  • Français

The Orderly Colored Longest Path Problem – a survey of applications and new algorithms

Published online by Cambridge University Press:  05 December 2013

Marta Szachniuk ,
Maria Cristina De Cola ,
Giovanni Felici  and
Jacek Blazewicz
Marta Szachniuk
Affiliation:
Institute of Bioorganic Chemistry, Polish Academy of Sciences, Noskowskiego 12/14, 61-704 Poznan, Poland. mszachniuk@cs.put.poznan.pl Institute of Computing Science, Poznan University of Technology, Piotrowo 2, 60-965 Poznan, Poland; mszachniuk@cs.put.poznan.pl; jblazewicz@cs.put.poznan.pl
Maria Cristina De Cola
Affiliation:
IRCCS Centro Neurolesi “Bonino Pulejo”, S.S. 113 Via Palermo c/da Casazza, 98123 Messina, Italy; cristina.decola@gmail.com
Giovanni Felici
Affiliation:
Institute of Systems Analysis and Computer Science “A. Ruberti”, National Research Council, V.le Manzoni 30, 00185 Rome, Italy; giovanni.felici@iasi.cnr.it
Jacek Blazewicz
Affiliation:
Institute of Bioorganic Chemistry, Polish Academy of Sciences, Noskowskiego 12/14, 61-704 Poznan, Poland. mszachniuk@cs.put.poznan.pl Institute of Computing Science, Poznan University of Technology, Piotrowo 2, 60-965 Poznan, Poland; mszachniuk@cs.put.poznan.pl; jblazewicz@cs.put.poznan.pl
Get access

Abstract

A concept of an Orderly Colored Longest Path (OCLP) refers to the problem of finding the longest path in a graph whose edges are colored with a given number of colors, under the constraint that the path follows a predefined order of colors. The problem has not been widely studied in the previous literature, especially for more than two colors in the color arrangement sequence. The recent and relevant application of OCLP is related to the interpretation of Nuclear Magnetic Resonance experiments for RNA molecules. Besides, an employment of this specific graph model can be found in transportation, games, and grid graphs. OCLP models the relationships between consecutive edges of the path, thus it appears very useful in representing the real problems with specific ties between their components. In the paper, we show OCLP’s correlation with similar issues known in graph theory. We describe the applications, three alternative models and new integer programming algorithms to solve OCLP. They are formulated by means of max flow problems in a directed graph with packing constraints over certain partitions of nodes. The methods are compared in a computational experiment run for a set of randomly generated instances.

Keywords

Edge colored graphlongest path problemalternating path
Type
Research Article
Information
RAIRO - Operations Research , Volume 48 , Issue 1 , January 2014 , pp. 25 - 51
Copyright
© EDP Sciences, ROADEF, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abouelaoualim, A., Das, K.Ch., Faria, L., Manoussakis, Y., Martinhon, C., Saad, R., Paths and trails in edge-colored graphs. Theor. Comput. Sci. 409 (2008) 497–510. Google Scholar
Adamiak, R.W., Blazewicz, J., Formanowicz, P., Gdaniec, Z., Kasprzak, M., Popenda, M., Szachniuk, M., An algorithm for an automatic NOE pathways analysis of 2D NMR spectra of RNA duplexes. J. Comput. Biol. 11 (2004) 163–179. Google ScholarPubMed
R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows: Theory, Algorithms and Applications. Prentice Hall, Englewood Cliffs, New Jersey (1993).
Bang-Jensen, J., Gutin, G., Alternating cycles and trails in 2-edge-colored complete multigraphs. Disc. Math. 188 (1998) 61–72. Google Scholar
Bang-Jensen, J., Gutin, G., Properly colored hamiltonian paths in edge-coloured complete graphs. Disc. Appl. Math. 82 (1998) 247–250. Google Scholar
B. Beauquier, S.Perennes and M. Syska, Efficient access to optical bandwidth routing and grooming in WDM networks: state-of-the-art survey. RESCCO Report IST-2001-33135, Universite de Nice-Sophia Antipolis (2002).
Benkouar, A., Manoussakis, Y.G., Paschos, V.Th. and Saad, R., On the complexity of some hamiltonian and eulerian problems in edge-colored complete graphs. Lect. Not. Comput. Sci. 557 (1991) 190–198. Google Scholar
D.P. Bertsekas, Network Optimization: Continuous and Discrete Models. Athena Scientific (1998).
J. Bialogrodzki, Path coloring and routing in graphs, in Contemporary Mathematics, edited by M. Kubale. American Mathematical Society (2004) 139–152.
J. Blazewicz, M. Szachniuk and A. Wojtowicz, Evolutionary approach to NOE paths assignment in RNA structure elucidation. Proceedings of the 2004 IEEE Symposium on Computational Intelligence in Bioinformatics and Computational Biology (2004) 206–213.
Blazewicz, J., Szachniuk, M., Wojtowicz, A., RNA tertiary structure determination: NOE pathways construction by tabu search. Bioinformatics 21 (2005) 2356–2361. Google ScholarPubMed
Blazewicz, J., Hammer, P.L., Lukasiak, P., Predicting secondary structures of proteins. IEEE Eng. Med. Biol. Mag. 24 (2005) 88–94. Google ScholarPubMed
B. Bollobás, Graph Theory. An Introductory Course, Springer-Verlag, New York (1979).
Bollobás, B., Erdos, P., Alternating hamiltonian cycles. Isr. J. Math. 23 (1976) 126–131. Google Scholar
J.A. Bondy, U.S.R Murty, Graph Theory with Applications, Macmillan Press, London (1976).
Carrabs, F., Cerulli, R., Gentili, M., The Labeled Maximum Matching Problem. Comput. Oper. Res. 36 (2009) 1859–1867. Google Scholar
R. Cerulli, A. Fink, M. Gentili, S. Voss, Metaheuristics comparison for the minimum labelling spanning tree problem, in The Next Wave on Computing, Optimization, and Decision Technologies, edited by B.L. Golden, S. Raghavan and E.A. Wasil. Springer, New York (2005) 93–106.
Cerulli, R., Fink, A., Gentili, M., Voss, S., Extensions of the Minimum Labelling Spanning Tree Problem. J. Telecom. Inform. Technol. 4 (2006) 39–45. Google Scholar
Cherkassky, B.V., Goldberg, A.V., Negative-cycle detection algorithms. Math. Prog. 85 (1999) 277–311. Google Scholar
Chou, W.S., Manoussakis, Y., Megalakaki, O., Spyratos, M., Tuza, Zs., Paths through fixed vertices in edge-colored graphs. Mathematiques, Informatique et Sciences Humaines 127 (1994) 49–58. Google Scholar
De Cola, M.C., Felici, G., Szachniuk, M., The Orderly Colored Longest Path Problem. CNR-IASI Technical Report 29 (2012). Google Scholar
Conrad, A., Hindrichs, T., Morsy, H., Wegener, I., Solution of the Knight’s Hamiltonian Path Problem on Chessboards. Disc. Appl. Math. 50 (1994) 125–134. Google Scholar
Dorninger, D., Hamiltonian circuits determining the order of chromosomes. Disc. Appl. Math. 50 (1994) 159–168. Google Scholar
Feng, J., Giesen, H.-E., Guo, Y., Gutin, G., Jensen, T., Rafiey, A., Characterization of edge-colored complete graphs with properly colored hamiltonian paths. J. Graph Theor. 53 (2006) 333–346. Google Scholar
P. Festa, The shortest path tour problem: problem definition, modeling, and optimization. Proceedings of INOC’2009 (2009) 1–7.
Gallo, G., Pallottino, S., Shortest path methods: a unifying approach. Math. Program. Stud. 26 (1986) 38–64. Google Scholar
M.R. Garey, D.S. Johnson, Computers and Intractability: A guide to the theory of NP-completeness. W.H. Freeman & Co., San Francisco CA (1979).
Gourvès, L., Lyra, A., Martinhon, C., Monnot, J., Protti, F., On s-t paths and trails in edge-colored graphs. Electron. Notes Discrete Math. 35 (2009) 221–226. Google Scholar
M. Kchikech, O. Togni, Paths coloring algorithms in mesh networks. Lect. Not. Comput. Sci. (2003) 193–202.
A. Kosowski, Path assignment with wavelength constraints. Ph.D. Thesis, Gdansk University of Technology, Poland (2006).
Li, H., Wang, G., Zhou, S., Long alternating cycles in edge-colored complete graphs. Lect. Not. Comput. Sci. 4613 (2007) 305–309. Google Scholar
F. Luccio, C. Mugnia, Hamiltonian paths on a rectangular chess-board. Proceedings of 16th Annual Allerton Conference (1978) 161–173.
Manoussaskis, Y., Alternating paths in edge-colored complete graphs. Disc. Appl. Math. 56 (1995) 297–309. Google Scholar
Myers, B.R., Enumeration of tour in hamiltonian rectangular lattice graphs. Math. Mag. 54 (1981) 19–23. Google Scholar
C. Nomikos, Coloring in Graphs, Ph.D. Thesis, Dept.of Electrical and Computer Engineering, National Technical University of Athens, Greece (1997).
Pevzner, P.A., DNA physical mapping and alternating eulerian cycles in colored graphs. Algorithmica 13 (1995) 77-105. CrossRefGoogle Scholar
P.A. Pevzner, Computational Molecular Biology: an algorithmic approach. MIT Press (2000).
Popenda, M., Szachniuk, M., Antczak, M., Purzycka, K.J., Lukasiak, P., Bartol, N., Blazewicz, J., Adamiak, R.W., Automated 3D structure composition for large RNAs. Nucleic Acids Res. 40 (2012) e112. Google ScholarPubMed
Popenda, M., Blazewicz, M., Szachniuk, M., Adamiak, R.W., RNA FRABASE version 1.0: an engine with a database to search for the three-dimensional fragments within RNA structures. Nucleic Acids Res. 36 (2008) D386-D391. Google ScholarPubMed
M. Popenda, M. Szachniuk, M. Blazewicz, S. Wasik, E.K. Burke, J. Blazewicz, R.W. Adamiak, RNA FRABASE 2.0: an advanced web-accessible database with the capacity to search the three-dimensional fragments within RNA structures. BMC Bioinformatics11 (2010) 231.
M. Szachniuk, M. Popenda, R.W. Adamiak, J. Blazewicz, An assignment walk through 3D NMR spectrum. Proceedings of the 2009 IEEE Symposium on Computational Intelligence in Bioinformatics and Computational Biology (2009) 215–219.
Szachniuk, M., Malaczynski, M., Pesch, E., Burke, E.K., Blazewicz, J., MLP accompanied beam search for the resonance assignment problem. J. Heuristics 19 (2013) 443–464. Google Scholar
M. Szachniuk, M.C. De Cola, G. Felici, D. de Werra, J. Blazewicz, Optimal pathway reconstruction on 3D NMR maps. Disc. Appl. Math. (2013) submitted for publication.
Szeider, S., Finding paths in graphs avoiding forbidden transitions. Disc. Appl. Math. 126 (2003) 239–251. Google Scholar
I.L. Tseng, H.W. Chen, C.I. Lee, Obstacle-aware longest path routing with parallel MILP solvers. Proceedings of WCECS (2010).
Wang, Y., Desmedt, Y., Edge-colored graphs with applications to homogeneous faults. Inf. Proc. Lett. 111 (2011) 634–641. Google Scholar
Watkins, J.J., Hoenigman, R.L., Knight’s tours on a torus. Mathematics 70 (1997) 175–184. Google Scholar
T. Zok, M. Popenda, M. Szachniuk, MCQ4Structures to compute similarity of molecule structures. Central Eur. J. Oper. Res. (2013), doi:10.1007/s10100-013-0296-5.
Yeo, A., A note on alternating cycles in edge-colored graphs. J. Comb. Theor. B 69 (1997) 222–225. Google Scholar
http://prolland.free.fr/works/research/dsat/dimacs.html.