The Parameterized Complexity of Cycle Packing: Indifference is Not an Issue R. Krithika 1,2 , Abhishek Sahu 1,2(B ) , Saket Saurabh 1,2,3 , and Meirav Zehavi 4 1 The Institute of Mathematical Sciences, HBNI, Chennai, India {rkrithika,asahu,saket}@imsc.res.in 2 UMI ReLax, Chennai, India 3 University of Bergen, Bergen, Norway 4 Ben-Gurion University, Beersheba, Israel meiravze@bgu.ac.il Abstract. 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. Several structural parameteriza- tions for Cycle Packing have been studied in the literature and our FPT algorithm fills a gap in the ecology of such 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. 1 Introduction Packing problems form a fundamental class of optimization problems in com- puter science. They involve finding a collection of objects with certain properties – examples include Bin Packing,Knapsack,Independent Set and Cycle Packing. Here, we focus on the Cycle Packing problem in the realm of param- eterized complexity. In the Cycle Packing problem, we are given an undirected graph G and a positive integer r, and the task is to check whether there exist r vertex-disjoint cycles. Since the publication of the classic Erd¨ os-P´osatheoremin 1965 [13], this problem has received significant scientific attention in the fields of Graph Theory and Algorithm Design. In particular, Cycle Packing is one of the first problems studied in the framework of parameterized complexity. In this framework, each problem instance is associated with a non-negative integer k called parameter, and a problem is said to be fixed-parameter tractable (FPT) if it can be solved in f (k)n O(1) time for some function f , where n is the input size. Due to space limitations, most proofs have been omitted. c  Springer International Publishing AG, part of Springer Nature 2018 M. A. Bender et al. (Eds.): LATIN 2018, LNCS 10807, pp. 712–726, 2018. https://doi.org/10.1007/978-3-319-77404-6_52