CMPUT 675: Approximation Algorithms and Approximability
Fall 2015, Tue and Thr 2:00-3:20:, in CSC B43.
Instructor: Mohammad R. Salavatipour

Purpose:
Most interesting optimization problems are NP-hard, and therefore it is unlikely that we can find optimal solutions for them efficiently. In many situations, finding a solution that is provably close to an optimal one is also acceptable. The next step is to show this is (almost) the best approximation one can hope for. These are the main goals of this course: find provably good approximation algorithms for the problems that are hard to solve exactly; and prove that finding better approximations are hard. We study some of the common and classical techniques in the design of approximation algorithms, followed by study of some more recent results in this field. Furthermore, we talk about the complexity of approximating these problems. This will be done by learning some classical and some more recent results on hardness of approximation.

Prerequisites:
CMPUT 304 or strong undergraduate background in theoretical computer science and mathematics. You must also have basic knowledge of graph theory.

Textbook:
There is no required text, but we will be using the following two books:

• The Design of Approximation Algorithms by David Williamson and David Shmoys, Cambridge University Press, 2011

• Approximation Algorithms by Vijay Vazirani, Springer-Verlag, Berlin, 2001.

• Design Techniques:
• Greedy and combinatorial methods,
  
• Dynamic programming and sclaing,
• Local search
• random sampling
  
• Linear programming (deterministic rounding, randomized rounding, primal-dual, iterative rounding)
• semi-definite programming and rounding
• metric embeddings
  
  
• Problems to discuss:
• Vertex Cover, Steiner tree. Set Cover
  
• TSP (symmetric and other variants)
• knapsack, bin packing
• Scheduling problems
• Facility location, k-median, k-center
  
• Steiner forest, Steiner network
• k-cut, multi-way cut, multi-cut, sparsest cut
  
• max-cut, max-sat
• Hardness of approximation (Intro to PCP, simple proofs, improved PCP's).
  

Lecture notes:

• Week 1: Introduction, Vertex Cover, Set Cover, Steiner Tree, TSP.
• Lecture 3: Introduction to LP, Set Cover via LP rounding,
  
• Lecture 4: Set Cover via Randomized rounding, Knapsack, Bin packing
• Lecture 5: Bin packing, Max SAT
  
• Lecture 6: Max SAT, Facility Location
  
• Lecture 7 & 8: Facility Location, K-Center, K-Median
  
• Lecture 9: Primal-Dual, Steiner Forest
• Lecture 10: Multiway Cut
• Lecture 11 & 12: Multiway cut, Multicut
• Lecture 13: Generalized Assignment Problem, MST polytope
  
• Lecture 14 (older notes): MST polytope, Min-cost Bounded Degree Spanning tree
• Lecture 15 & 16: Iterative Rounding (Min-cost Bounded degree Spanning tree, Survivable Network Design)
• Lecture 17 & 18: Semidefinite programing (Max-cut, Max-2SAT), Metric embeddings
  
• Lecture 19 & 20: Approximation of metrics by tree metrics (older notes), PTAS for Euclidean TSP (older notes)
  
• Lecture 20 & 21: Intro. to Hardness of approximation, PCP (Older notes)
• Lecture 22 & 23: Label Cover, Hardness of Set cover (older notes)
  

Template for course notes Here is a sample (and in PDF) and its source file.
Here is the algorithms.sty

Assignments:

• Assignment 1
• Assignment 2
• Assignment 3
  

Grading policy:
This is a theory course (no programming involved). There will be 4 take home assignments;
Also, each participant in the course is required to provide scribe notes for one or two weeks of lectures. This is worth 10% of your final mark.
Scribe notes for each week are due the next Monday at noon. Scribe notes must be typed in LaTeX using the template provided above.

Useful Links:
Here are some useful links to more resources (books, course notes by other people who have taught this course, problems, etc.):

• Links to similar courses offered by:
  Moses Charikar, Chandra Chekuri, Joseph Cheriyan, Anupam Gupta and R. Ravi, David Williamson,
  

• Notes on linear programming by Michel X. Goemans ,

• Approximation Algorithms for NP-hard Problems. Dorit Hochbaum (Ed.), PWS Publishers, 1996. Below are links to some of the Chapters of this book that are available online: Hardness of Approximations by Sanjeev Arora and Carsten Lund, Approximation Algorithms for Bin Packing: A Survey by E.G. Coffman, M.R. Garey, and D.S. Johnson , Cut Problems and their application to divide-and-conquer, by David Shmoys . All of these are part of the book "Approximation Algorithms for NP-hard problems" listed above. Copyrights for the material are held by PWS Publishing with all rights reserved.

• Text on Computational Complexity:
  Sanjeev Arora and Boaz Barak, Complexity Theory: A Modern Approach. ( homepage).

• R. Motwani and P. Raghavan, Randomized algorithms, Cambridge University Press, Cambridge, 1995.

• A compendium of NP optimization problems , by Pierluigi Crescenzi and Viggo Kann.

Questions? Send email to me ...