Paintshop, odd cycles and necklace splitting

International audienceThe 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

A topological characterization of modulo-p arguments and implications for necklace splitting

The classes PPA-p have attracted attention lately, because they are the main candidates for capturing the complexity of Necklace Splitting with p thieves, for prime p. However, these classes were not known to have complete problems of a topological nature, which impedes any progress towards settling the complexity of the Necklace Splitting problem. On the contrary, topological problems have been pivotal in obtaining completeness results for PPAD and PPA, such as the PPAD-completeness of finding a Nash equilibrium [18, 15] and the PPA-completeness of Necklace Splitting with 2 thieves [24]. In this paper, we provide the first topological characterization of the classes PPA-p. First, we show that the computational problem associated with a simple generalization of Tucker's Lemma, termed p-polygon-Tucker, as well as the associated Borsuk-Ulam-type theorem, p-polygon-Borsuk-Ulam, are PPA-p-complete. Then, we show that the computational version of the well-known BSS Theorem [8], as well as the associated BSS-Tucker problem are PPA-p-complete. Finally, using a different generalization of Tucker's Lemma (termed Zp-star-Tucker), which we prove to be PPA-p-complete, we prove that p-thief Necklace Splitting is in PPA-p. This latter result gives a new combinatorial proof for the Necklace Splitting theorem, the only proof of this nature other than that of Meunier [42]. All of our containment results are obtained through a new combinatorial proof for Zp-versions of Tucker's lemma that is a natural generalization of the standard combinatorial proof of Tucker's lemma by Freund and Todd [27]. We believe that this new proof technique is of independent interest