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On Minimum Monotone and Unimodal Partitions of Permutations

Stefano, Gabriele Di; Krause, Stefan; LĂĽbbecke, Marco E.; Zimmermann, Uwe T.

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

Partitioning a permutation into a minimum number of monotone subsequences is NP-hard. We extend this complexity result to minimum partitions into unimodal subsequences. In graph theoretical terms these problems are cocoloring and what we call split-coloring of permutation graphs. Based on a network flow interpretation of both problems we introduce mixed integer programs; this is the first approach to obtain optimal partitions for these problems in general. We derive an LP rounding algorithm which is a 2-approximation for both coloring problems. It performs much better in practice. In an online situation the permutation becomes known to an algorithm sequentially, and we give a logarithmic lower bound on the competitive ratio and analyze two online algorithms.