Paintshop, odd cycles and necklace splitting

The following problem is well-known under various forms from car manufacturing and has been presented in Discrete Applied Mathematics(136) by T.Epping, W.Hochstättler and P.Oertel as a combinatorial optimization problem: 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 above paper states two conjectures about the general problem. In this paper we identify the problem concerning two colors as a well-known problem in graphs, and deduce the polynomial solvability of some paintshop scheduling problems, and provide new simple proofs of the NP-completeness and APX-completeness of the problem in general. The main result we present in this direction states that such binary clutters have a transversal that contains at most 3/8-th of the elements (ignoring those that are forced to be in the transversal for some obvious reason), and such a transversal can be found in polynomial time. We then recognize the coincidence of the two aforementioned conjectures with wellknown and difficult problems of combinatorial topology: general necklace splitting and a well-identified special case of necklace splitting. These two problems have been solved by Alon, Goldberg and West, and their necklace splitting theorems confirm the validity of both conjectures. Briefly, polynomial algorithms, minmax theorems, tight bounds and NP-completeness results are deduced for paintshop scheduling problems making clear what can be solved, what cannot and what is still open. 1 Problem formulation In [7], T.Epping, W.Hochstättler and P.Oertel introduced the following problem. The origins of the model lie in car manufacturing with individual demands, which is reported to occur often in Europe. Given a sequence of cars where repetition can occur, and for each car a multiset of colors where the sum of the multiplicities is equal to the number of repetitions of the car in the sequence, decide the color to be applied for each occurrence of each car so that each color occurs with the multiplicity that has been assigned. The goal is to minimize the number of color changes in the sequence. If cars are considered to be letters in an alphabet, the following is a formalization. ∗LVMT, Ecole Nationale des Ponts et Chaussées, 6-8 avenue Blaise Pascal, Cité Descartes Champs-surMarne, 77455 Marne-la-Vallée cedex 2, France. E-mail: [email protected] †CNRS, research partially supported my the Marie Curie network “ADONET” of the European Community, E-mail: [email protected]