Abstract
The celebrated Erdős-Pósa theorem states that every undirected graph that does not admit a family of k vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size \(\cal{O}(k\log k)\). The analogous result for directed graphs has been proven by Reed, Robertson, Seymour, and Thomas, but their proof yields a nonelementary dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing.
We show that we can obtain a polynomial bound if we relax the disjointness condition. More precisely, we show that if in a directed graph G there is no family of k cycles such that every vertex of G is in at most two (resp. four) of the cycles, then there exists a feedback vertex set in G of size \(\cal{O}(k^{6})\) (resp. \(\cal{O}(k^{4})\)). We show also variants of the above statements for butterfly minor models of any strongly connected digraph that is a minor of a directed cylindrical grid and for quarter-integral packings of subgraphs of high directed treewidth.
Similar content being viewed by others
References
S. A. Amiri, K. Kawarabayashi, S. Kreutzer and P. Wollan: The ErdŐs-Pósa property for directed graphs, CoRR, arXiv:1603.025042016.
M. Andrews, J. Chuzhoy, V. Guruswami, S. Khanna, K. Talwar and L. Zhang: Inapproximability of edge-disjoint paths and low congestion routing on undirected graphs, Combinatorica 30 (2010), 485–520.
T. Carpenter, A. Salmasi and A. Sidiropoulos: Routing symmetric demands in directed minor-free graphs with constant congestion, CoRR, abs/1711.01692, 2017.
C. Chekuri and J. Chuzhoy: Large-treewidth graph decompositions and applications, In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC 2013), pages 291–300. ACM, 2013.
C. Chekuri and J. Chuzhoy: Polynomial bounds for the grid-minor theorem, Journal of the ACM, 63 (2016), 40:1–40:65.
C. Chekuri and A. Ene: The all-or-nothing flow problem in directed graphs with symmetric demand pairs, Mathematical Programming, 1–24, 2014.
C. Chekuri, A. Ene and M. Pilipczuk: Constant congestion routing of symmetric demands in planar directed graphs, SIAM Journal on Discrete Mathematics 32 (2018), 2134–2160.
C. Chekuri, S. Khanna and F. Shepherd: Multicommodity flow, well-linked terminals, and routing problems, in: Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC 2005), 183–192. ACM, 2005.
J. Chuzhoy and S. Li: A polylogarithmic approximation algorithm for edge-disjoint paths with congestion 2, Journal of the ACM 63 (2016), 1–51.
J. Chuzhoy and Z. Tan: Towards tight(er) bounds for the excluded grid theorem, Journal of Combinatorial Theory, Series B 146 (2021), 219–265.
E. D. Demaine and M. Hajiaghayi: Linearity of grid minors in treewidth with applications through bidimensionality, Combinatorica 28 (2008), 19–36.
P. ErdŐs and L. Pósa: On independent circuits contained in a graph, Canadian Journal of Mathematics 17 (1965), 347–352.
M. Hatzel, K. Kawarabayashi and S. Kreutzer: Polynomial planar directed grid theorem, in: Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019), 1465–1484, 2019.
T. Johnson, N. Robertson, P. D. Seymour and R. Thomas: Directed tree-width, Journal of Combinatorial Theory, Series B 82 (2001), 138–154.
T. Johnson, N. Robertson, P. D. Seymour and R. Thomas: Excluding a grid minor in planar digraphs, CoRR, arXiv:1510.00473, 2001.
K. Kawarabayashi and S. Kreutzer: The directed grid theorem, CoRR, arXiv:1411.5681v1, 2014.
K. Kawarabayashi and S. Kreutzer: The directed grid theorem, in: Proceedings of the 47th Annual ACM on Symposium on Theory of Computing (STOC 2015), 655–664, 2015.
S. Kreutzer and S. Ordyniak: Width-measures for directed graphs and algorithmic applications, in: Quantitative Graph Theory: Mathematical Foundations and Applications, Springer, 2014.
T. Masařík, I. Muzi, M. Pilipczuk, P. Rzążewski and M. Sorge: Packing Directed Circuits Quarter-Integrally, in: 27th Annual European Symposium on Algorithms (ESA 2019), volume 144 of LIPIcs, pages 1–13, Dagstuhl, Germany, 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
B. Reed: Introducing directed tree width, Electronic Notes in Discrete Mathematics 3 (1999), 222–229.
B. Reed, N. Robertson, P. Seymour and R. Thomas: Packing directed circuits, Combinatorica 16 (1996), 535–554.
B. A. Reed and D. R. Wood: Polynomial treewidth forces a large grid-like-minor, Eur. J. Comb. 33 (2012), 374–379.
N. Robertson and P. D. Seymour: Graph minors. III. Planar tree-width, J. Comb. Theory, Ser. B 36 (1984), 49–64.
N. Robertson and P. D. Seymour: Graph minors. V. Excluding a planar graph, J. Comb. Theory, Ser. B 41 (1986), 92–114.
N. Robertson, P. D. Seymour and R. Thomas: Quickly excluding a planar graph, J. Comb. Theory, Ser. B 62 (1994), 323–348.
P. D. Seymour: Packing directed circuits fractionally, Combinatorica 15 (1995), 281–288.
Acknowledgments
We thank Stephan Kreutzer (TU Berlin) for interesting discussions on the topic and for pointing out Lemma 4. We also thank an anonymous reviewer for pointing out Theorem 7 as a corollary of our results.
This research is part of projects that have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme Grant Agreements 648527 (Irene Muzi) and 714704 (all authors).
Tomáš Masařík was also supported by student grant number SVV-2017-260452 of Charles University, Prague, Czech Republic.
Author information
Authors and Affiliations
Corresponding author
Additional information
An extended abstract of this manuscript appeared at European Symposium on Algorithms 2019 [19].
Rights and permissions
About this article
Cite this article
Masařík, T., Muzi, I., Pilipczuk, M. et al. Packing Directed Cycles Quarter- and Half-Integrally. Combinatorica 42 (Suppl 2), 1409–1438 (2022). https://doi.org/10.1007/s00493-021-4743-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-021-4743-y