Hostname: page-component-cb9f654ff-5kfdg Total loading time: 0 Render date: 2025-09-07T15:43:57.323Z Has data issue: false hasContentIssue false
  • English
  • Français

Constructive Packings by Linear Hypergraphs

Published online by Cambridge University Press:  12 September 2013

JILL DIZONA  and
BRENDAN NAGLE
JILL DIZONA
Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA (e-mail: jdizona@mail.usf.edu, bnagle@usf.edu)
BRENDAN NAGLE
Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA (e-mail: jdizona@mail.usf.edu, bnagle@usf.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

For k-graphs F0 and H, an F0-packing of H is a family $\mathscr{F}$ of pairwise edge-disjoint copies of F0 in H. Let νF0(H) denote the maximum size |$\mathscr{F}$| of an F0-packing of H. Already in the case of graphs, computing νF0(H) is NP-hard for most fixed F0 (Dor and Tarsi [6]).

In this paper, we consider the case when F0 is a fixed linear k-graph. We establish an algorithm which, for ζ > 0 and a given k-graph H, constructs in time polynomial in |V(H)| an F0-packing of H of size at least νF0(H) − ζ |V(H)|k. Our result extends one of Haxell and Rödl, who established the analogous algorithm for graphs.

Keywords

Primary 05C35Secondary 05C6505C85

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 6 , November 2013 , pp. 829 - 858
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Alon, N. and Spencer, J. (1992) The Probabilistic Method, Wiley Interscience.Google Scholar
[2]Alon, N., Duke, R., Lefmann, H., Rödl, V. and Yuster, R. (1994) The algorithmic aspects of the Regularity Lemma. J. Algorithms 16 80–109.CrossRefGoogle Scholar
[3]Conlon, D., Hàn, H., Person, Y. and Schacht, M. (2012) Weak quasi-randomness for uniform hypergraphs. Random Struct. Alg. 40 1–38.CrossRefGoogle Scholar
[4]Czygrinow, A. Personal communication.Google Scholar
[5]Czygrinow, A. and Rödl, V. (2000) An algorithmic regularity lemma for hypergraphs. SIAM J. Comput. 30 1041–1066.CrossRefGoogle Scholar
[6]Dor, D. and Tarsi, M. (1992) Graph decomposition is NPC: A complete proof of Holyer's conjecture. In Proc. 20th ACM STOC, ACM Press, pp. 252–263.Google Scholar
[7]Frankl, P. and Rödl, V. (1992) The uniformity lemma for hypergraphs. Graphs Combin. 8 309–312.CrossRefGoogle Scholar
[8]Frankl, P. and Rödl, V. (2002) Extremal problems on set systems. Random Struct. Alg. 20 131–164.CrossRefGoogle Scholar
[9]Gowers, W. T. (2006) Quasirandomness, counting and regularity for 3-uniform hypergraphs. Combin. Probab. Comput. 15 143–184.CrossRefGoogle Scholar
[10]Gowers, W. T. (2007) Hypergraph regularity and the multidimensional Szemerédi theorem. Ann. of Math. (2) 166 897–946.CrossRefGoogle Scholar
[11]Grable, D. (1996) Nearly-perfect hypergraph packing is in NC. Inform. Process. Lett. 60 295–299.CrossRefGoogle Scholar
[12]Haxell, P. E. and Rödl, V. (2001) Integer and fractional packings in dense graphs. Combinatorica 21 13–38.CrossRefGoogle Scholar
[13]Haxell, P. E., Nagle, B. and Rödl, V. (2003) Integer and fractional packings in dense 3-uniform hypergraphs. Random Struct. Alg. 22 248–310.CrossRefGoogle Scholar
[14]Haxell, P. E., Nagle, B. and Rödl, V. (2005) An algorithmic version of the hypergraph regularity method [extended abstract]. In Proc. 46th Annual IEEE Symposium on Foundations of Computer Science: FOCS '05, pp. 439–448.CrossRefGoogle Scholar
[15]Haxell, P. E., Nagle, B. and Rödl, V. (2008) An algorithmic version of the hypergraph regularity method. SIAM J. Comput. 37 1728–1776.CrossRefGoogle Scholar
[16]Janson, S., Luczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.CrossRefGoogle Scholar
[17]Kohayakawa, Y., Nagle, B., Rödl, V. and Schacht, M. (2010) Weak hypergraph regularity and linear hypergraphs. J. Combin. Theory Ser. B 100 151–160.CrossRefGoogle Scholar
[18]Kohayakawa, Y., Rödl, V. and Thoma, L. (2003) An optimal algorithm for checking regularity. SIAM J. Comput. 32 1210–1235.CrossRefGoogle Scholar
[19]Nagle, B., Poerschke, A., Rödl, V. and Schacht, M. (2009) Hypergraph regularity and quasirandomness. In Proc. 20th Annual ACM–SIAM Symposium on Discrete Algorithms: SODA '09 (Mathieu, C., ed.), ACM Press, pp. 227–245.Google Scholar
[20]Rödl, V. and Skokan, J. (2004) Regularity lemma for uniform hypergraphs. Random Struct. Alg. 25 1–42.CrossRefGoogle Scholar
[21]Rödl, V., Schacht, M., Siggers, M. H. and Tokushige, N. (2007) Integer and fractional packings of hypergraphs. J. Combin. Theory Ser. B 97 245–268CrossRefGoogle Scholar
[22]Skokan, J. and Thoma, L. (2004) Bipartite subgraphs and quasi-randomness. Graphs Combin. 20 255–262.CrossRefGoogle Scholar
[23]Szemerédi, E. (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arithmetica 27 199–245.CrossRefGoogle Scholar
[24]Szemerédi, E. (1978) Regular partitions of graphs. In Problèmes en Combinatoires et Théorie des Graphes: Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976, CNRS, pp. 399–401.Google Scholar
[25]Williams, V. V. (2012) Multiplying matrices faster than Coppersmith–Winograd. In Proc. 44th Symposium on Theory of Computing: STOC '12, ACM Press, pp. 887–898.CrossRefGoogle Scholar
[26]Yuster, R. (2005) Integer and fractional packings of families of graphs. Random Struct. Alg 26 110–118.CrossRefGoogle Scholar