Skip to main content Accessibility help
×
Home

A natural barrier in random greedy hypergraph matching

  • Patrick Bennett (a1) and Tom Bohman (a2)

Abstract

Let r ⩾ 2 be a fixed constant and let $ {\cal H} $ be an r-uniform, D-regular hypergraph on N vertices. Assume further that D → ∞ as N → ∞ and that degrees of pairs of vertices in $ {\cal H} $ are at most L where L = D/( log N)ω(1). We consider the random greedy algorithm for forming a matching in $ {\cal H} $ . We choose a matching at random by iteratively choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with a chosen edge before moving on to the next choice. This process terminates when there are no edges remaining in the graph. We show that with high probability the proportion of vertices of $ {\cal H} $ that are not saturated by the final matching is at most (L/D)(1/(2(r−1)))+o(1). This point is a natural barrier in the analysis of the random greedy hypergraph matching process.

Copyright

Corresponding author

*Corresponding author. Email: tbohman@math.cmu.edu

Footnotes

Hide All

Research supported in part by NSF grant DMS-1001638 and Simons Foundation grant #426894.

Research supported in part by NSF grants DMS-1001638 and DMS-1100215.

Footnotes

References

Hide All
[1]Alon, N., Kim, J. and Spencer, J. (1997) Nearly perfect matchings in regular simple hypergraphs. Israel J. Math. 100 171187.
[2]Bohman, T. (2009) The triangle-free process. Adv. Math. 221 16531677.
[3]Bohman, T., Frieze, A. and Lubetzky, E. (2010) A note on the random greedy triangle packing algorithm. J. Combin. 1 477488.
[4]Bohman, T., Frieze, A. and Lubetzky, E. (2015) Random triangle removal. Adv. Math. 280 379438.
[5]Bohman, T. and Picollelli, M. (2012) Evolution of SIR epidemics on random graphs with a fixed degree sequence. Random Struct. Alg. 41 179214.
[6]Erd˝os, P. and Hanani, H. (1963) On a limit theorem in combinatorial analysis. Publ. Math. Debrecen 10 1013.
[7]Grable, D. (1997) On random greedy triangle packing. Electron. J. Combin. 4 R11.
[8]Kostochka, A. and Rödl, V. (1998) Partial Steiner systems and matchings in hypergraphs. Random Struct. Alg. 13 335347.
[9]Pippenger, N. and Spencer, J. (1989) Asymptotic behavior of the chromatic index for hypergraphs. J. Combin. Theory Ser. A 51 2442.
[10]Rödl, V. (1985) On a packing and covering problem. European J. Combin. 6 6978.
[11]Rödl, V. and Thoma, L. (1996) Asymptotic packing and the random greedy algorithm. Random Struct. Alg. 8 161177.
[12]Spencer, J. (1995) Asymptotic packing via a branching process. Random Struct. Alg. 7 167172.
[13]Telcs, A., Wormald, N. and Zhou, S. (2007) Hamiltonicity of random graphs produced by 2-processes. Random Struct. Alg. 31 450481.
[14]Vu, V. (2000) New bounds on nearly perfect matchings in hypergraphs: Higher codegrees do help. Random Struct. Alg. 17 2963.
[15]Wormald, N. (1999) The differential equation method for random graph processes and greedy algorithms. In Lectures on Approximation and Randomized Algorithms (Karonski, M. and Prömel, H. J., eds), PWN, pp. 73155.

MSC classification

A natural barrier in random greedy hypergraph matching

  • Patrick Bennett (a1) and Tom Bohman (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed