Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-17T12:19:46.382Z Has data issue: false hasContentIssue false
  • English
  • Français

Orientability Thresholds for Random Hypergraphs

Part of: Graph theory

Published online by Cambridge University Press:  23 March 2015

PU GAO  and
NICHOLAS WORMALD
PU GAO
Affiliation:
Max-Planck-Institut für Informatik, 66123 Saarbrücken, Saarland, Germany and Department of Combinatorics and Optimization, University of Waterloo, Canada (e-mail: [email protected])
NICHOLAS WORMALD
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Canada (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let h > w > 0 be two fixed integers. Let H be a random hypergraph whose hyperedges are all of cardinality h. To w-orient a hyperedge, we assign exactly w of its vertices positive signs with respect to the hyperedge, and the rest negative signs. A (w,k)-orientation of H consists of a w-orientation of all hyperedges of H, such that each vertex receives at most k positive signs from its incident hyperedges. When k is large enough, we determine the threshold of the existence of a (w,k)-orientation of a random hypergraph. The (w,k)-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, when h = 2 and w = 1, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran. This settled a conjecture of Karp and Saks.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C85: Graph algorithms
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Issue 5 , September 2015 , pp. 774 - 824
Copyright
Copyright © Cambridge University Press 2015 

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.)

References

REFERENCES

[1] Alon, N. and Spencer, J. (2008) The Probabilistic Method, third edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.CrossRefGoogle Scholar
[2] Azar, Y., Broder, A. Z., Karlin, A. R. and Upfal, E. (1999) Balanced allocations. SIAM J. Comput. 29 180–200.CrossRefGoogle Scholar
[3] Berenbrink, P., Czumaj, A., Steger, A. and Vőcking, B. (2000) Balanced allocations: The heavily loaded case. In 32nd Annual ACM Symposium on Theory of Computing, pp. 745–754.Google Scholar
[4] Bollobás, B. and Frieze, A. M. (1985) On matchings and Hamiltonian cycles in random graphs. In Random Graphs '83, Vol. 118 of North-Holland Mathematics Studies, North-Holland, pp. 23–46.Google Scholar
[5] Cain, J., Sanders, P. and Wormald, N. (2007) The random graph threshold for k-orientability and a fast algorithm for optimal multiple-choice allocation. In Symposium on Discrete Algorithms: SODA, pp. 469–476.Google Scholar
[6] Cain, J. and Wormald, N. (2006) Encores on cores. Electron. J. Combin. 13 RP81.Google Scholar
[7] Chernoff, H. (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 493–507.Google Scholar
[8] Chvátal, V. (1991) Almost all graphs with 1.44n edges are 3-colorable. Random Struct. Alg. 2 11–28.CrossRefGoogle Scholar
[9] Cook, W. J., Cunningham, W. H., Pulleyblank, W. R. and Schrijver, A. (1998) Combinatorial Optimization, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.Google Scholar
[10] Dietzfelbinger, M., Goerdt, A., Mitzenmacher, M., Montanari, A., Pagh, R. and Rink, M. (2012) Tight thresholds for cuckoo hashing via XORSAT. Automata, Languages and Programming, Lecture Notes in Computer Science 6198 213–225.Google Scholar
[11] Dietzfelbinger, M. and Meyer auf der Heide, F. (1993) Simple, efficient shared memory simulations. In 5th ACM Symposium on Parallel Algorithms and Architectures, pp. 110–119.Google Scholar
[12] Fernholz, D. and Ramachandran, V. (2007) The k-orientability thresholds for Gn,p . In Proc. ACM–SIAM Symposium on Discrete Algorithms: SODA, pp. 459–468.Google Scholar
[13] Fountoulakis, N. and Panagiotou, K. (2012) Sharp load thresholds for cuckoo hashing. Random Structures & Algorithms 41 306–333.CrossRefGoogle Scholar
[14] Frieze, A. and Melsted, P. (2012) Maximum matchings in random bipartite graphs and the space utilization of cuckoo hashtables. Random Structures & Algorithms 41 334–364.Google Scholar
[15] Gao, P. Generation and properties of random graphs and analysis of randomized algorithms. PhD dissertation, Department of Combinatorics and Optimization, University of Waterloo. http://hdl.handle.net/10012/4987 Google Scholar
[16] Gao, P. and Wormald, N. C. (2010) Load balancing and orientability thresholds for random hypergraphs. In STOC'10, Cambridge, Massachusetts, USA. pp. 97–103.Google Scholar
[17] Hakimi, S. L. (1965) On the degrees of the vertices of a directed graph. J. Franklin Inst. 279 290–308.Google Scholar
[18] Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.Google Scholar
[19] Karp, R. M., Luby, M. and Meyer auf der Heide, F. (1992) Efficient PRAM simulation on a distributed memory machine. In 24th ACM Symposium on Theory of Computing, pp. 318–326.Google Scholar
[20] McKay, B. D. and Wormald, N. C. (1991) Asymptotic enumeration by degree sequence of graphs with degrees o(n 1/2). Combinatorica 11 369–382.Google Scholar
[21] Meyer auf der Heide, F., Scheideler, C. and Stemann, V. (1996) Exploiting storage redundancy to speed up randomized shared memory simulations. Theoret. Comput. Sci. 162 245–281.CrossRefGoogle Scholar
[22] Mitzenmacher, M., Richa, A. and Sitaraman, R. (2001) The power of two random choices: A survey of techniques and results. Chapter 9 of Handbook of Randomized Computing, Vol. I, Combinatorial Optimization, Kluwer, pp. 255–312.Google Scholar
[23] Pittel, B., Spencer, J. and Wormald, N. (1996) Sudden emergence of a giant k-core in a random graph. J. Combin. Theory Ser. B 67 111–151.CrossRefGoogle Scholar
[24] Pittel, B. and Wormald, N. (2003) Asymptotic enumeration of sparse graphs with a minimum degree constraint. J. Combin. Theory Ser. A 101 249–263.Google Scholar
[25] Sanders, P., Egner, S. and Korst, J. (2003) Fast concurrent access to parallel disks. Algorithmica 35 21–55.Google Scholar
[26] Schrijver, A. (2003) Combinatorial Optimization: Polyhedra and Efficiency, Vol. B, Vol. 24 of Algorithms and Combinatorics, Springer.Google Scholar
[27] Wormald, N. C. (1995) Differential equations for random processes and random graphs. Ann. Appl. Probab. 5 1217–1235.CrossRefGoogle Scholar
[28] Wormald, N. C. (1999) The differential equation method for random graph processes and greedy algorithms. Lectures on Approximation and Randomized Algorithms (Karonski, M. and Prömel, H. J., eds), PWN, Warsaw, pp. 73–155.Google Scholar
[29] Wormald, N. C. (2004) Random graphs and asymptotics. Section 8.2 of Handbook of Graph Theory (Gross, J. L. and Yellen, J., eds), CRC, pp. 817–836.Google Scholar