hal-00835238
https://enpc.hal.science/hal-00835238
doi:10.1016/j.dam.2008.06.017
[SHS] Sciences de l'Homme et de la Société
[ENPC] Ecole des Ponts ParisTech
[UGA] HAL Grenoble Alpes
[CNRS] CNRS - Centre national de la recherche scientifique
[UNIV-GRENOBLE1] Université Joseph Fourier - Grenoble I
[INPG] Institut polytechnique de Grenoble
[G-SCOP] LABORATOIRE G-SCOP
[AO-SOCIOLOGIE] Archives ouvertes de la Sociologie
[SOCIOLOGIE] SOCIOLOGIE
[PARISTECH] ParisTech
[ENPC-LVMT] Laboratoire Ville, Mobilité, Transport
[G-SCOP_OSP_OC] Optimisation Combinatoire
[G-SCOP_OSP] Optimisation des systèmes de Production
[IFSTTAR] Ifsttar
[TDS-MACS] Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes
[UNIV-EIFFEL] Université Gustave Eiffel
[IFSTTAR-UNIVEIFFEL] Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux
[UPEM-UNIVEIFFEL] Université Paris-Est Marne-la-Vallée
Paintshop, odd cycles and necklace splitting
Meunier, Frédéric
Sebő, András
[SHS.SOCIO] Humanities and Social Sciences/Sociology
[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
[INFO.INFO-RO] Computer Science [cs]/Operations Research [cs.RO]
ART
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The following problem has been presented in [T. Epping, W. Hochstättler, P. Oertel, Complexity results on a paint shop problem, Discrete Applied Mathematics 136 (2004) 217-226] by Epping, Hochstättler and Oertel: cars have to be painted in two colors in a sequence where each car occurs twice; assign the two colors to the two occurrences of each car so as to minimize the number of color changes. More generally, the "paint shop scheduling problem" is defined with an arbitrary multiset of colors given for each car, where this multiset has the same size as the number of occurrences of the car; the mentioned article states two conjectures about the general problem and proves its NP-hardness. In a subsequent paper in [P. Bonsma, Th. Epping, W. Hochstättler, Complexity results for restricted instances of a paint shop problem for words, Discrete Applied Mathematics 154 (2006) 1335-1343], Bonsma, Epping and Hochstättler proved its APX-hardness and noticed the applicability of some classical results in special cases. We first identify the problem concerning two colors as a minimum odd circuit cover problem in particular graphs, exactly situating the problem. A resulting two-way reduction to a special minimum uncut problem leads to polynomial algorithms for subproblems, to observing APX-hardness through MAX CUT in 3-regular graphs, and to a solution with at most 3/4th of all possible remaining color changes (when all obliged color changes have been made). For the general problem concerning an arbitrary number of colors, we realize that the two aforementioned conjectures are corollaries of the celebrated "necklace splitting" theorem of Alon, Goldberg and West. © 2008 Elsevier B.V. All rights reserved.
2009
en
Discrete Applied Mathematics
Elsevier