Packing Cycles Faster Than Erdos--Posa

Abstract

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 2 O(k 2 ) · |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 2 O(k log 2 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 2 o(k log 2 k) · |V | O(1) , beating the bound 2 O(k log 2 k) · |V | O(1) , has been found. In light of this, it seems natural to ask whether the 2 O(k log 2 k) · |V | O(1) bound is essentially optimal. In this paper, we answer this question negatively by developing a 2 O( k log 2 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 2 O(k log 2 k) · |V | O(1) , our algorithm runs in time linear in |V |, and its space complexity is polynomial in the input size.

Packing Arc-Disjoint Cycles in Tournaments

“…The classical Erdős-Pósa theorem for cycles in undirected graphs states that for each nonnegative integer k, every undirected graph either contains k vertex-disjoint cycles or has a feedback vertex set consisting of f (k) = O(k log k) vertices [21]. An interesting consequence of this theorem is that it leads to an FPT algorithm for Vertex-Disjoint Cycle Packing (see [36] for more details). Analogous to these results, we prove an Erdős-Pósa type theorem for tournaments and show that it leads to an O (2 O(k log k) ) time algorithm and a linear vertex kernel for ACT.…”

“…Vertex-Disjoint Cycle Packing in undirected graphs is FPT with respect to the solution size k [10,36] but has no polynomial kernel unless NP ⊆ coNP/poly [11]. In contrast, Edge-Disjoint Cycle Packing in undirected graphs admits a kernel with O(k log k) vertices (and is therefore FPT) [11].…”

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

“…No polynomial time O(log 1/2−ε n)-approximation is possible for this problem [118] for any ε > 0, unless every problem in NP can be solved in randomized quasi-polynomial time. Furthermore, despite being FPT [119] parameterized by the solution size, VERTEX CYCLE PACKING does not admit any polynomial-sized exact kernel for this parameter [120], unless NP⊆coNP/poly. Nevertheless, a PSAKS can be found [18].…”

“…Furthermore, for some problems including CYCLE PACKING, the Erdős-Pósa property gives an immediate parameterized algorithm. We refer the reader to a recent survey [139] and papers [119,140,141].…”

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

“…No polynomial time O(log 1/2−ε n)-approximation is possible for this problem [107] for any ε > 0, unless every problem in NP can be solved in randomized quasi-polynomial time. Furthermore, despite being FPT [108] parameterized by the solution size, VERTEX CYCLE PACKING does not admit any polynomial-sized exact kernel for this parameter [109], unless NP⊆coNP/poly. Nevertheless, a PSAKS can be found [18].…”