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

¹ Institute of Bioorganic Chemistry, Polish Academy of Sciences, Noskowskiego 12/14, 61-704 Poznan, Poland
mszachniuk@cs.put.poznan.pl
² Institute of Computing Science, Poznan University of Technology, Piotrowo 2, 60-965 Poznan, Poland
mszachniuk@cs.put.poznan.pl; jblazewicz@cs.put.poznan.pl
³ IRCCS Centro Neurolesi “Bonino Pulejo”, S.S. 113 Via Palermo c/da Casazza, 98123 Messina, Italy
cristina.decola@gmail.com
⁴ Institute of Systems Analysis and Computer Science “A. Ruberti”, National Research Council, V.le Manzoni 30, 00185 Rome, Italy
giovanni.felici@iasi.cnr.it

Received: 21 September 2013
Accepted: 30 September 2013

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.