Fixed-Parameter Algorithms for Cochromatic Number and Disjoint Rectangle Stabbing

Abstract

Given a permutation π of {1,...,n} and a positive integer k, we give an algorithm with running time $2^{O(k^2 \log k)}n^{O(1)}$ that decides whether π can be partitioned into at most k increasing or decreasing subsequences. Thus we resolve affirmatively the open question of whether the problem is fixed parameter tractable. This NP-complete problem is equivalent to deciding whether the cochromatic number (the minimum number of cliques and independent sets the vertices of the graph can be partitioned into) of a given permutation graph on n vertices is at most k. In fact, we give a more general result: within the mentioned running time, one can decide 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, namely greedy localization and iterative compression. We further demonstrate the power of this combination by giving a $2^{O(k^2 \log k)}n \log n$ time algorithm for deciding whether a given set of n non-overlapping axis-parallel rectangles can be stabbed by at most k of the given set of horizontal and vertical lines. Whether such an algorithm exists was mentioned as an open question in several papers.

This work is supported by the Research Council of Norway.