Packing Cycles Faster Than Erd\H{o}s-P\'osa

The Cycle Packing problem asks whether a given undirected graph G = (V,E) contains k vertex-disjoint cycles. Since the publication of the classic Erdős-Pósa theorem in 1965, this problem received significant scientific attention in the fields of Graph Theory and Algorithm Design. In particular, this problem is one of the first problems studied in the framework of Parameterized Complexity. The non-uniform fixed-parameter tractability of Cycle Packing follows from the Robertson–Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time 2O(k) · |V | using exponential space. In case a solution exists, Bodlaender’s algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a 2O(k log2 k) · |V |-time (deterministic) algorithm using exponential space, which is a consequence of the Erdős-Pósa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on Parameterized Complexity. Yet, no algorithm that runs in time 2o(k log2 k) · |V |O(1), beating the bound 2O(k log2 k) · |V |O(1), has been found. In light of this, it seems natural to ask whether the 2O(k log2 k) · |V |O(1) bound is essentially optimal. In this paper, we answer this question negatively by developing a 2O( k log2 k log log k ) · |V |-time (deterministic) algorithm for Cycle Packing. In case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the bound 2O(k log2 k) · |V |O(1), our algorithm runs in time linear in |V |, and its space complexity is polynomial in the input size. 1998 ACM Subject Classification G.2.2 Graph Algorithms, I.1.2 Analysis of Algorithms