CME 305: Discrete Mathematics and Algorithms

Topics Covered

• Basic Algebraic Graph Theory
• Minimum Spanning Trees and Matroids
• Maximum Flow and Submodularity
• NP-Hardness
• Approximation Algorithms
• Randomized Algorithms
• The Probabilistic Method
• Spectral Sparsification

Course Description

This course is targeting doctorate students with strong foundations in mathematics who wish to become more familiar with the design and analysis of discrete algorithms. An undergraduate course in algorithms is not a prerequisite, only familiarity with basic notions in linear algebra and discrete mathematics.

Required textbook:
Algorithm Design by Kleinberg and Tardos [KT]

Optional textbooks:
Graph Theory by Reinhard Diestel [D]
Approximation Algorithms by Vijay Vazirani [V]
Randomized Algorithms by Rajeev Motwani and Prabakhar Raghavan [MR]
The Probabilistic Method by Noga Alon and Joel Spencer [AS]

Midterm: Thursday Feb. 13 in class
Final: Wednesday Mar. 19, 7-10PM in 200-205 (in the quad, NOT Sequoia)

Assignments

• Assignment 1. Solutions. Due at the beginning of class Thursday 01/23/14
• Assignment 2. Solutions. Due at the beginning of class Thursday 02/06/14
• Assignment 3. Solutions. Due at the beginning of class Thursday 02/27/14
• Assignment 4. Solutions. Due at the beginning of class Thursday 03/13/14

References

• Tu 1/7: Lecture 1 (Intro to Dynamic Programming): KT 6.1-6.4
• Th 1/9: Lecture 2 (Trees, Eulerian Circuits, MST): KT 3.1, 4.5; Notes
• Tu 1/14: Lecture 3 (Min-Cut, Max-Flow, Ford-Fulkerson): KT 7; Notes
• Th 1/16: Lecture 4 (s-t Min-Cut/Max-Flow Applications, Global Min-Cut): Previous Notes; Notes; A New Approach to the Minimum Cut Problem (Karger and Stein)
• Tu 1/21: Lecture 5 (Randomized Algorithms): KT 13.1-13.5; MR 1, 6
• Th 1/23: Lecture 6 (Randomized Walks and Electrical Networks): Notes; MR 6 (Highly Recommended)
• Tu 1/28: Lecture 7 (Concentration Inequalities): MR 3-4, 6.5; KT 13.9
• Th 1/30: Lecture 8 (The Probabilistic Method): Notes; MR 5; AS 1
• Tu 2/4: Lecture 9 (Problem Classes, Reductions): Notes; The NP Compendium; KT 8
• Th 2/6: Lecture 10 (Approximation Algorithms): Notes; KT 11
• Tu 2/11: Lecture 11 (Midterm Review)
• Th 2/13: Lecture 12 (Midterm)
• Tu 2/18: Lecture 13 (Goemans-Williamson Max-Cut, Bin Packing): V 26; CMU Max-Cut Notes; Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming (Goemans and Williamson)
• Th 2/20: Lecture 14 (Asymmetric TSP, Spectral Graph Theory, Sparsification Intro): An O(log n/log log n)-approximation Algorithm for ATSP (Asadpour et al.); Sparsification Notes
• Tu 2/25: Lecture 15 (Graph Sparsification): Sparsification Notes; Graph Sparsification by Effective Resistances (Spielman and Srivastava)
• Th 2/27: Lecture 16 (Graph Sparsification): Sparsification Notes; Graph Sparsification by Effective Resistances (Spielman and Srivastava)
• Tu 3/4: Lecture 17 (Matroids): MIT Matroid Notes
• Th 3/6: Lecture 18 (Applications: ML and Clustering): An Impossibility Theorem for Clustering (Kleinberg); A Uniqueness Theorem for Clustering (Zadeh and Ben-David)
• Tu 3/11: Lecture 19 (Applications: Distributed Computing): NodeIterator Analysis; Counting Triangles with Spark
• Th 3/13: Lecture 20 Exam Review

Exams

Midterm Solutions pdf,

2011 Midterm pdf, Solutions pdf.

2010 Midterm pdf, Solutions pdf.

Practice Midterm 1 (In-Class) pdf, Solutions pdf.

Practice Midterm 2 (In-Class) pdf, Solutions pdf.

Spring 2012 Qual pdf.

Fall 2012 Qual pdf.

Spring 2013 Qual pdf.

Problem Sessions

Problem Session 2/10/14 and Solutions

Problem Session 2/20/14 and Solutions

Pre-Final Workshop