Computer Science and Engineering
SUNY at Buffalo

### CSE 694: Topics in Algorithms - Probabilistic Analysis and Randomized Algorithms

Instructor: Hung Q. Ngo

### Fall 2008

T-Th, 9:30-10:50am
Capen 260 (map)

There will be (hopefully light) changes to the following schedule throughout the semester. Please check back regularly!
Week Topics Notable Events
1. Aug 26 A Gentle Introduction to Probability Theory, Probabilistic Method, and Randomized Algorithms
• Reading: Chapers 1 to 4
• Lecture notes 1 in PDF. Concepts: Discrete Probability Spaces, the Union Bound, the Probabilistic Method. Examples: Sperner's Lemma, Ramsey Numbers, d-disjunct Matrices
• Lecture notes 2 in PDF: Conditional Probability, Independence, Randomized Algorithms, Random Variables, Expectation and its Linearity, Conditional Expectation, Law of Total Probability.
• Lecture notes 3 in PDF: Concepts: (Co)Variance, Coupon Collector Problem, Concentration/Tail Inequalities (Markov, Chebyshev, Chernoff). Examples: Probabilistic Packet Marking, Sampling and Estimation, Randomized Routing on the Hypercube.

Concepts:
• Probability Spaces and Events
• Independence and Conditional Probability
• Random variables, Expectation, Linearity of Expectation, Variance
• Conditional Distributions, Conditional Expectation
• Basic distributions: Bernoulli, Binomial, Geometric
• Derandomization with the method of conditional probability
• Moments
• Tail bounds: Markov, Chebyshev, Chernoff
• Sample mean, median, sampling and estimation
Examples:
• Extremal Combinatorics
• Ramsey Numbers
• Sperner Lemma
• Randomized Algorithms:
• Randomized Min-Cut
• Randomized Quicksort Analsysis
• Max-E3SAT
• Networking:
• Probabilistic Packet Marking
• Permuation routing on the Hypercube
• Coding Theory and Group Testing:
• Expander code
• Disjunct matrices

Tue Aug 26
- Homework 1 out. Elementary practice exercises on discrete probabilities and the probabilistic method
2. Sep 02

3. Sep 09 Tue, Sep 09
- Homework 1 due.
- Homework 2 out
4. Sep 16 The Probabilistic Method
• Lecture notes 4 in PDF. Concepts: the union bound technique. Examples: "nice" tournaments, 2-coloring of uniform hypergraphs, expander codes.
• Lecture notes 5 in PDF. Concepts: the argument from expectation. Exampes: large cuts in graphs, linear combinations of unit vectors, unbalancing lights.
• Lecture notes 6 in PDF. Concepts: the alteration technique. Examples: Independent Sets, Dominating Sets, 2-coloring of uniform hypergraphs (revisited).
• Lecture notes 7 in PDF. Concepts: the second moment method, the G(n,p) random graph model. Examples: distinct subset sums, 4-clique property
• Lecture notes 8 in PDF. Concepts: Lovasz local lemma. Examples: hypergraph coloring (again), k-SAT, edge-disjoint paths.

Concepts

• Union bound
• Argument from expectation
• Alterations
• Second moment method; also the G(n,p) random graph model
• Lovasz local lemma
Examples
• Tournaments
• 2-coloring of uniform hypergraphs
• Expander codes
• Finding large cuts, Max SAT
• Linear combinations of unit vectors
• Unbalancing lights
• Dominating sets
• Independent Sets
• Distinct subset sums
• The G(n,p) random graph model and 4-clique property
• Edge-disjoint paths
• k-SAT

5. Sep 23
6. Sep 30

Tue Sep 30: No Class. Rosh Hashanah

Thu, Oct 02:
- Homework 3 due.
- Homework 4 out

7. Oct 07 Thu, Oct 09: No Class. Yom Kippur
8. Oct 14
Randomized Algorithms
• Reading: Chaper 7, 9, Part of Chapter 11
• Lecture notes 9 in PDF. Concepts: linear programming (LP), integer linear programming (ILP), duality, complementary slackness. Examples: maxflow, mincut, multiway cut, set cover, vertex cover, max-SAT.
• Lecture notes 10 in PDF. Concepts: randomized rounding. Example: maxflow-mincut theorem, multiway cut.
• Lecture notes 11 in PDF. Concepts: randomized rounding. Examples: weighted set cover.
• Lecture notes 12 in PDF. Concepts: randomized rounding, derandomization. Examples: satisfiability problems.
• Lecture notes 13 in PDF. Concepts: semidefinite programming, randomized rounding. Example: max-cut. See also the following notes for more on Semidefinite programming and Max-Cut.
• Laci Lovasz, "Semidefinite programs and combinatorial optimization", [ postscript ]
• Uriel Feige, "The use of semidefinite programming in approximation algorithms", [ ppt ]
• Uri Zwick, "Semidefinite programming-based approximation algorithms", [ ppt ]
• M. Goemans, "MAXCUT, SDP-based 0.878-approximation algorithm", [ pdf ]
• Prasad Raghavendra, "Optimal Algorithms and Inapproximability Results for Every CSP?", [ paper in pdf | presentation in ppt ]
• Lecture x in PDF: Introduction to data streams (if time allows! Well, it looks like we'll have to skip this, along with approximate sampling and counting.)

Concepts

• Approximation Algorithms for Combinatorial Optimization Problems
• Linear Programming, Integer Programming
• Semidefinite programming
• Randomized Rounding
• Basic Monte Carlo Method
• Complexity of Counting (#P, #P-Completeness)
• Approximate Counting
• "Equivalence" between Approximate Counting and Uniform Sampling
• Random walks on graphs, Eigenvalue Connection, Expanders
• Saving Random Bits, Derandomization
Examples
• randomized rounding
• satisfiability problems
• covering problems
• cut problems
• randomized algorithm for s,t-connectivity in log-space
• randomized algorithm for 2-sat
• saving random bits
• #DNF
• 01-Permanent
• Network Reliability
• (May be) Volume Estimation
• Data Streams:
• Frequency Moment Estimation
• Hot Item Estimation

9. Oct 21
10. Oct 28

Tue, Oct 28:
- Homework 3 due.
- Homework 4 out

11. Nov 04
12. Nov 11 Computational Learning Theory
Thu, Nov 13:
- Homework 4 due.
- Homework 5 out
13. Nov 18
14. Nov 25 Thu, Nov 27: No Class. Thanks Giving!
15. Dec 02 Thu, Dec 04:
- Homework 5 due.

Friday Dec 05 is the last day of classes.

16. Dec 09

DONE!