On minimum k-modal partitions of permutations

Gabriele Di Stefano, Stefan Krause, Marco E. Lübbecke, Uwe T. Zimmermann
2008 Journal of Discrete Algorithms  
Partitioning a permutation into a minimum number of monotone subsequences is N P-hard. We extend this complexity result to minimum partitioning into k-modal subsequences; here unimodal is the special case k = 1. Based on a network flow interpretation we formulate both, the monotone and the k-modal version, as mixed integer programs. This is the first proposal to obtain provably optimal partitions of permutations. LP rounding gives a 2-approximation for minimum monotone partitions and a (k +
more » ... proximation for minimum (upper) k-modal partitions. For the online problem, in which the permutation becomes known to an algorithm sequentially, we derive a logarithmic lower bound on the competitive ratio for minimum monotone partitions, and we analyze two (bin packing) online algorithms. These immediately apply to online cocoloring of permutation graphs.
doi:10.1016/j.jda.2008.01.002 fatcat:xpik3ui5znfxbcux2tje2pmd3m