Packing Cycles faster than Erdos-Pósa

Daniel Lokshtanov, Amer E. Mouawad, Saket Saurabh, Meirav Zehavi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 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.

Original languageAmerican English
Title of host publication44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
EditorsAnca Muscholl, Piotr Indyk, Fabian Kuhn, Ioannis Chatzigiannakis
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770415
DOIs
StatePublished - 1 Jul 2017
Externally publishedYes
Event44th International Colloquium on Automata, Languages, and Programming, ICALP 2017 - Warsaw, Poland
Duration: 10 Jul 201714 Jul 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume80

Conference

Conference44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
Country/TerritoryPoland
CityWarsaw
Period10/07/1714/07/17

Keywords

  • Cycle Packing
  • Erdos-Pósa theorem
  • Graph algorithms
  • Parameterized complexity

All Science Journal Classification (ASJC) codes

  • Software

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