The Parameterized Complexity of Cycle Packing: Indifference is Not an Issue

In the Cycle Packing problem, we are given an undirected graph G, a positive integer r, and the task is to check whether there exist r vertex-disjoint cycles. In this paper, we study Cycle Packing with respect to a structural parameter, namely, distance to proper interval graphs (indifference graphs). In particular, we show that Cycle Packing is fixed-parameter tractable (FPT) when parameterized by t, the size of a proper interval deletion set. For this purpose, we design an algorithm with O(2 ^{O ( t log t )}n^{O ( 1 )}) running time. Bodlaender et al. (Theor Comput Sci 511:117–136, 2013) studied several structural parameterizations for Cycle Packing and our FPT algorithm fills a gap in their ecology of parameterizations. We combine color coding, greedy strategy and dynamic programming based on structural properties of proper interval graphs in a non-trivial fashion to obtain the FPT algorithm. Our belief is that this approach is quite general and can be useful in solving many other problems with the same parameterization.