On Minimum k-Modal Partitions of Permutations

Abstract

Partitioning a permutation into a minimum number of monotone subsequences is ${\mathcal NP}$-hard. We extend this complexity result to minimum partitioning into k-modal subsequences, that is, subsequences having at most k internal extrema. 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. From these models we derive an LP rounding algorithm which is a 2-approximation for minimum monotone partitions and a (k+1)-approximation for minimum (upper) k-modal partitions in general; this is the first approximation algorithm for this problem. In computational experiments we see that the rounding algorithm performs even better in practice. For the associated 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 findings immediately apply to online cocoloring of permutation graphs; they are the first results concerning online algorithms for this graph theoretical interpretation.