The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
Approximating minimum cocolorings
Information Processing Letters
Developments from a June 1996 seminar on Online algorithms: the state of the art
Developments from a June 1996 seminar on Online algorithms: the state of the art
Selected Topics in Column Generation
Operations Research
On minimum k-modal partitions of permutations
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
Untangled Monotonic Chains and Adaptive Range Search
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Range queries over untangled chains
SPIRE'10 Proceedings of the 17th international conference on String processing and information retrieval
Untangled monotonic chains and adaptive range search
Theoretical Computer Science
On the online track assignment problem
Discrete Applied Mathematics
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Partitioning a permutation into a minimum number of monotone subsequences is NP-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+1)-approximation 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.