Fixed-parameter algorithms for Cochromatic Number and Disjoint Rectangle Stabbing via iterative localization

摘要

Given a permutation π of (1 ....,n) and a positive integer k, can 7t 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 log 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 log k))n log n 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.