TY - GEN
T1 - Packing Cycles faster than Erdos-Pósa
AU - Lokshtanov, Daniel
AU - Mouawad, Amer E.
AU - Saurabh, Saket
AU - Zehavi, Meirav
N1 - Publisher Copyright: © Daniel Lokshtanov, Amer E. Mouawad, Saket Saurabh, and Meirav Zehavi.
PY - 2017/7/1
Y1 - 2017/7/1
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 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(κ2) · |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(κ log2 κ) · |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(κ log2 κ) · |V |O(1), beating the bound 2O(κlog2κ) · |V |O(1), has been found. In light of this, it seems natural to ask whether the 2O(κ log2 κ) · |V |O(1) bound is essentially optimal. In this paper, we answer this question negatively by developing a 2O(κlog2κ/log log κ ) · |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(κ log2 κ) · |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 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(κ2) · |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(κ log2 κ) · |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(κ log2 κ) · |V |O(1), beating the bound 2O(κlog2κ) · |V |O(1), has been found. In light of this, it seems natural to ask whether the 2O(κ log2 κ) · |V |O(1) bound is essentially optimal. In this paper, we answer this question negatively by developing a 2O(κlog2κ/log log κ ) · |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(κ log2 κ) · |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=85027278281&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.ICALP.2017.71
DO - https://doi.org/10.4230/LIPIcs.ICALP.2017.71
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
A2 - Muscholl, Anca
A2 - Indyk, Piotr
A2 - Kuhn, Fabian
A2 - Chatzigiannakis, Ioannis
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
Y2 - 10 July 2017 through 14 July 2017
ER -