# Combinatorics and Graph Theory in Computer Science (Spring 2015)

Time and Location: TTh 12-1:15pm, Shaffer 304.

Instructor: Xin Li. Office hours: T 2:30-3:30pm, or by appointment.

Syllabus

Course description: This is a graduate level course studying the applications of combinatorics and graph theory in computer science. We will start with some basic combinatorial techniques such as counting and pigeon hole principle, and then move to advanced techniques such as the probabilistic method, spectral graph theory and additive combinatorics. We shall see their applications in various areas in computer science, such as proving lower bounds in computational models, randomized algorithms, coding theory and pseudorandomness.

Pre Requisite: Discrete math. Probability theory and linear algebra highly recommended.

Required Textbook: Stasys Jukna, Extremal Combinatorics: With Applications in Computer Science. 2nd Edition.
Errata

Optional Textbook: N. Alon and J. H. Spencer, "The Probabilistic Method"

List of Tentative Topics:
Basic Techniques:
Counting; Pigeon hole principle and resolution refutation lower bound; Matching and Hall's theorem.

The Probabilistic Method:
Basic method; Lovaz local lemma and its constructive proof; Linearity of Expectation; The deletion method; The entropy function; Random walks and randomized algorithm for CNF formulas

Spectral Graph theory:
Basic properties of graph spectrum; Cheeger's inequality and approximation of graph expansion; Expander graphs and applications to superconcentrators and pseudorandomness; Error correcting codes and expander codes; Small set expansion, Unique Games Conjecture and Hardness of approximation.

Sum product theorem, Szemeredi-Trotter theorem, Kakeya set problem and applications to randomness extractors.

Homework 1

Papers for potential projects:

Robin Moser and Gabor Tardos. A constructive proof of the general lovĂˇsz local lemma.

James R. Lee, Shayan Oveis Gharan and Luca Trevisan. Multi-way spectral partitioning and higher-order Cheeger inequalities.

Shayan Oveis Gharan and Luca Trevisan. Approximating the Expansion Profile and Almost Optimal Local Graph Clustering.

Adam Marcus, Daniel A. Spielman and Nikhil Srivastava. Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees.

Joshua Batson, Daniel A. Spielman and Nikhil Srivastava. Twice-Ramanujan Sparsifiers.

Klim Efremenko. 3-query locally decodable codes of subexponential length.

Zeev Dvir, Parikshit Gopalan and Sergey Yekhanin. Matching vector codes.

A. Bhowmick, Z. Dvir, and S. Lovett. New Bounds for Matching Vector Families.