TY - JOUR

T1 - Packing cycles faster than Erdos–Posa

AU - Lokshtanov, Daniel

AU - Mouawad, Amer E.

AU - Saurabh, Saket

AU - Zehavi, Meirav

N1 - Funding Information:
∗Received by the editors October 2, 2017; accepted for publication (in revised form) May 6, 2019; published electronically July 2, 2019. A preliminary version of this paper appeared in Proceedings of ICALP’17. https://doi.org/10.1137/17M1150037 Funding: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant 306992. †University of Bergen, Norway (daniello@ii.uib.no). ‡American University of Beirut, Lebanon (aa368@aub.edu.lb). §University of Bergen, Norway, and Institute of Mathmatical Sciences, HBNI, India (saket@imsc. res.in). ¶Ben-Gurion University, Israel (meiravze@bgu.ac.il).
Funding Information:
The research leading to these results has received funding from the European Research Council under the European Union?s Seventh Framework Programme (FP7/2007-2013)/ERC grant 306992. We would like to thank the reviewers for several suggestions and insightful remarks that have improved the presentation of the paper.
Publisher Copyright:
c 2019 Society for Industrial and Applied Mathematics

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The Cycle Packing problem asks whether a given undirected graph G = (V, E) contains k vertex-disjoint cycles. Since the publication of the classic Erdos–Pósa theorem in 1965, this problem received significant 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 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(k 2) · |V | using exponential space. In the 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 Erdos–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 (log k log log 2 k k ) · |V |-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) · |V |O (1), our algorithm runs in time linear in |V |, and its space complexity is polynomial in the input size.

AB - The Cycle Packing problem asks whether a given undirected graph G = (V, E) contains k vertex-disjoint cycles. Since the publication of the classic Erdos–Pósa theorem in 1965, this problem received significant 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 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(k 2) · |V | using exponential space. In the 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 Erdos–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 (log k log log 2 k k ) · |V |-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) · |V |O (1), our algorithm runs in time linear in |V |, and its space complexity is polynomial in the input size.

KW - Cycle packing

KW - Erdos–Pósa theorem

KW - Graph algorithms

KW - Parameterized complexity

UR - http://www.scopus.com/inward/record.url?scp=85071488539&partnerID=8YFLogxK

U2 - 10.1137/17M1150037

DO - 10.1137/17M1150037

M3 - Article

AN - SCOPUS:85071488539

VL - 33

SP - 1194

EP - 1215

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 3

ER -