PACKING CYCLES FASTER THAN ERDOS-POSA.

The Cycle Packing problem asks whether a given undirected graph G = (V;E) con- tains k vertex-disjoint cycles. Since the publication of the classic Erd}os{Posa theorem in 1965, this problem received significant attention in the fields of graph theory and algorithm design. In particu- lar, this problem is one of the first problems studied in the framework of parameterized complexity. The nonuniform 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(k2) jV j using exponential space. In the case a solu- tion 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) jV j-time (deterministic) algorithm using exponential space, which is a consequence of the Erd}os{Posa 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) jV jO(1), beating the bound 2O(k log2 k) jV jO(1), has been found. In light of this, it seems natural to ask whether the 2O(k log2 k) jV jO(1) bound is essentially optimal. In this paper, we answer this question negatively by developing a 2 O(k log2 k log log k) jV j-time (deterministic) algorithm for Cycle Packing. In the case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the bound 2O(k log2 k) jV jO(1), our algorithm runs in time linear in jV j, and its space complexity is polynomial in the input size. [ABSTRACT FROM AUTHOR]