Georgia Tech: Randomized Algorithms and Approximation Algorithms, Fall 1997 (CS 8113J)

Instructors: Leonard Schulman, Vijay Vazirani

Description:

This course is the first of a two-quarter sequence on Randomized Algorithms. This quarter, we will have a particular focus on Approximation Algorithms, an area in which randomness plays a significant role. The course is suitable for students to whom either area is new.

In the approximation portion, we plan on covering some of the most exciting current topics in LP-duality based methods, as well (topics 2 and 3 below). For this reason, students who took Vazirani's Approximation Algorithms course in the Spring quarter will also benefit from this course.

Prerequisites are some experience with algorithms (preferably at the basic graduate level, though less may suffice in some circumstances); and probability (non measure theoretic, at a good undergraduate or basic graduate level).

The following topics will be covered (time permitting):

1. Conductance, expanders, rapidly mixing Markov chains; coupling and canonical path arguments, and their applications to random generation and approximate counting of spanning trees and perfect matchings respectively.
2. Primal-Dual algorithms: for several problems including generalized Steiner network problem and k-MST problem.
3. Multicommodity flow: approximate max-flow min-cut theorems, approximating sparsest cuts and multicuts, embedding metrics in Euclidean space with low distortion (Bourgain's Theorem).
4. Yao's Lemma (von Neumann minimax), and applications.
5. Lovasz Local Lemma, Beck/Alon algorithm.
6. Hardness of approximability: MAX SNP, PCP.
7. Approximately counting DNF satisfying assignments and estimating network reliability.
8. Schwartz's Lemma and its applications to testing polynomial identities, computing matchings, and associativity testing.

Scribe notes of Schulman's lectures:

Lecture of 6 Oct 97 (Game tree evaluation)

Lecture of 8 Oct 97 (Randomized and distributional complexity)

Postscript to lecture of 8 Oct 97

Lecture of 29 Oct 97 (Hash functions)

Lecture of 3 Nov 97 (Statistical estimators; Chebyshev and Chernoff bounds; multiplicative approximation)

Lecture of 5 Nov 97 (Statistical estimators; estimating the permanent)

Lecture of 10 Nov 97 (Estimating the permanent)

Lecture of 12-17 Nov 97 (Verifying matrix multiplication; perfect matching in bipartite graphs; Schwartz-Zippel; Verifying associativity)

Lecture of 1 Dec 97 (Rapidly mixing Markov chains. Mixing time. Equivalence of approximate sampling and approximate counting for self-reducible problems.)

Lecture of 3 Dec 97 (Rapidly mixing Markov chains. Coupling. Uniformly sampling spanning trees.)