Given a permutation @p of {1,...,n} and a positive integer k, can @p be partitioned into at most k subsequences, each of which is either increasing or decreasing? We give an algorithm with running time 2^O^(^k^^^2^l^o^g^k^)n^O^(^1^) that solves this problem, thereby showing that it is fixed parameter tractable. This NP-complete problem is equivalent to deciding whether the cochromatic number of a given permutation graph on n vertices is at most k. Our algorithm solves in fact a more general problem: within the mentioned running time, it decides whether the cochromatic number of a given perfect graph on n vertices is at most k. To obtain our result we use a combination of two well-known techniques within parameterized algorithms: iterative compression and greedy localization. Consequently we name this combination ''iterative localization''. We further demonstrate the power of this combination by giving an algorithm with running time 2^O^(^k^^^2^l^o^g^k^)nlogn that decides whether a given set of n non-overlapping axis-parallel rectangles can be stabbed by at most k of a given set of horizontal and vertical lines.
['Pinar Heggernes', 'Dieter Kratsch', 'Daniel Lokshtanov', 'Venkatesh Raman', 'Saket Saurabh']
Fixed-parameter algorithms for Cochromatic Number and Disjoint Rectangle Stabbing via iterative localization