21-737 Probabilistic Combinatorics

This course covers the probabilistic method for combinatorics in detail and introduces randomized algorithms and the theory of random graphs.

Methods covered include the second moment method, the R\"odl nibble, the Lov\'asz local lemma, correlation inequalities, martingale's and tight concentration, Janson's inequality, branching processes, coupling and the differential equations method for discrete random processes. Objects studied include the configuration model for random regular graphs, Markov chains, the phase transition in the Erd\H{o}s-R\'enyi random graph, and the Barab\'asi-Albert preferential attachment model.

Course Instructor:

Tom Bohman
Wean Hall 6105
tbohman@math.cmu.edu
Office Hours: Wednesday 3:00-4:0 or by appointment


Course information: Postscript PDF

Homework 1: Postscript PDF
Hints for Homework 1: Postscript PDF

Homework 2: Postscript PDF
Hints for Homework 2: Postscript PDF

Homework 3: Postscript PDF
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Homework 4: Postscript PDF
Hints for Homework 4: Postscript PDF


Schedule of paper presentations:

Wednesday April 22

Friday, April 24

Monday, April 27

Wednesday, April 29

Friday, May 1


Tentative schedule:

Week 1: Alon and Spencer, Probabilistic Lenses `Triangle-free graphs have large independence numbers' and `Crossing numbers, incidences, sums and products'.

Week 2: Second moment method. See Alon and Spencer Sections 4.5 Clique Number and 4.6 Distinct Sums.

Week 3: Second moment method. Satisfiability threshold for random 2-SAT. See Chvatal and Reed: Mick gets some (the odds are on his side), FOCS, 1992, pp. 620--627.

Week 4: Lovasz Local Lemma (and linear arboricity). Alon and Spencer: Sections 5.1, 5.2, 5.5

Week 5: The Moser-Tardos Local Lemma Algorithm. See Moser and Trados: A constructive proof of the general Lovász Local Lemma, Journal of the ACM 57 (2010) (2) Art. 11, 15pp. Also see some notes written by Joel Spencer.

Week 6: Janson's Inequality. See Alon and Spencer Section 8.1-8.3. Diameter of a random graph.

Week 7: Martingales and tight concentration. See Alon and Spencer 7.1-7.4.

Week 8: An Introduction to Dynamic Concentration

Week 9: Branching processes and the giant component. See Janson, Luczak and Rucinski Section 5.2

Week 10: Applictaions of Dynamic concentration. Giant Component Triangle Free Process

Week 11: Introduction to finite markov chains. See Chapters 1, 2 of Levin, Peres, and Wilmer.

Week 12: Markov chain monte carlo and mixing. See Chapters 3, 4, 5 of Levin, Peres, and Wilmer.

Week 13: Path coupling. See Chapter 14 of Levin, Peres, and Wilmer.

Week 14,15: Student presentations.


Possible presentation papers:

D. Achlioptas and C. Moore, The Asymptotic Order of the k-SAT Threshold, Proc. Foundations of Computer Science (FOCS) 2002.

Y. Azar, A. Broder, A. Karlin, and E. Upfal, Balanced allocations, SIAM J. on Computing, 29, (2000), 180-200.

T. Bohman, A. Frieze and E. Lubetzky, A note on the random greedy triangle-packing algorithm , Journal of Combinatorics, 1 (2010), 477-488.

Svante Janson, The probability that a random multigraph is simple. Combin. Probab. Comput. 18 (2009), 205-225.

M. Krivelevich, Bounding Ramsey numbers through large deviation inequalities, Random Structures and Algorithms 7 (1995), 145-155.

A Nachmias, Y. Peres, The critical random graph, with martingales, Israel Journal of Math, 176 (2010) 29-43.

A Nachmias, Y. Peres, Component sizes of the random graph outside the scaling window , Latin American Journal of Probability and Mathematical Statistics (ALEA), 3, 133-142 (2007).

B. Reed and B. Sudakov, Asymptotically the list colouring constants are 1, J. Combinatorial Theory Ser. B 86 (2002), 27-37.

A. Rucinski and N. Wormald, Random graph processes with degree restrictions, Combinatorics, Probability and Computing 1 (1992) 169-180.

J. Spencer Asymptotic Packing via A Branching Process Random Structures and Algorithms, 7 (1995,) 167-172

J. Spencer and N. Wormald, Birth control for giants , Combinatorica 27 (2007), 587-628.