Schedule

Course details can be found . This schedule is tentative and subject to change, particularly toward the last half of the quarter (as I usually end up covering less than I hope). When a topic is not listed, we will continue where we left off in the previous class.

Week Tuesday Thursday Other
1 Introduction: analyses besides optimality
Vijay Vazirani: 1, 3.1, 3.2
Christofides' algorithm (1976)
Vijay Vazirani: 3.1, 3.2
Linear programming
Vijay Vazirani: 12
Finally beating Christofides' algorithm:
(1.5-eps)-approx, winter 2010
1.461-approx, spring 2011
(13/9)-approx, summer 2011
``` 2
```
LP-based approximation algorithms
Vijay Vazirani: 14 (and possibly 13.1)
Problem solving session A
Prepare at least 3 different algorithms/analyses that give a 2-approximation for vertex cover. (See problems 1.3 and 14.7 for inspiration.)
Write-up for problem solving session A
Write up a 2-approx for vertex cover

(state the algorithm and analysis of approx ratio)

Jeff Erickson: LP and duality
I will try very hard to comment on

written solutions by the next class
after they are handed in.

3

Approximation schemes
Vijay Vazirani: 8
Approximation schemes continued
Vijay Vazirani: 11
On the oddly unsettled sum-of-square-roots problem
Slightly out-of-date details on the hardness of approximation of metric-TSP
``` 4
```
Problem solving session B
Exercise 8.3 in the text.
Proof of the bound on the nth Catalan number.
Explain why Catalan numbers bound the number

of non-crossing patterns for TSP
Write-up for problem solving session B
Give a pseudopolynomial-time algorithm for the subset-sum ratio problem, complete with proof of correctness

Parameterized complexity for PTAS design
Jeff Erickson: Treewidth
Section 11.3 to be covered in more detail the following week.
An example of a primal-dual algorithm where
the problem of interest is the maximization problem.
``` 5
```
Tree decompositions cont. Approximation schemes for planar graphs
An application of dominating set to error-correcting codes.
``` 6
```
A guide to polynomial-time approximation schemes for planar network design problems
Problem solving session C
(update) Prove: given a planar graph of degree d with spanning tree of the dual with longest simple path having p edges has carving-width p+d-1.
``` 7
```

Online and competitive analyses
Luca Trevisan: lecture 17

rent-or-buy notes Poly-log competitive ratio for k-server
``` 8
```
Problem solving session D
Problem 3
Max-cut
Vijay Vazirani: Chapter 26.0 - 26.4
Ellipsoid method notes
``` 9
```
Stable and planted analyses
Tim Roughgarden: lecture 3
``` 10
```
Problem solving session E
Vazirani Question 2.3 (max k-cut) and Question 26.12 (constrained max cut)
There may be at most one more question
``` 11
```
Final exam week
Any remaining assignments will be graded on June 15. Please hand in any remaining submissions by the end of the previous day, June 14. You may slip these under my office door.
Enjoy the summer!

Course Information

This course on advanced algorithms will cover approximation algorithms and randomized algorithms and perhaps other topics to be determined. We will be using the textbook Approximation Algorithms by Vazirani (corrected 2nd printing, 2003), Tim Roughgarden's lecture notes and Luca Trevisan's lecture notes. Other (free) materials may be provided throughout the course.

There will be two main components to this course, and you will be graded as such:

• Participating
• Discussion of textbook and lecture-note material: It is expected that you will read this material before class, but not necessarily fully understand the material. We will use lecture time to go through the material in depth and solve related problems.
• Discussion of problem solutions: It is expected that you will come to class with a solution to the assigned problem. Or a partial solution. Or questions that highlight where you got stuck or need help. You are highly encouraged to work together to solve these problems. The goal during class is to make sure that everyone understands the solution to these problems.
• Evaluation: How helpful you are in helping the class solve problems and understand material? You will evaluate yourself and your classmates once a week. Participation is important both inside and outside class. If your participation is above average, your grade will be moved up a level (e.g. B+ to A-) or two (e.g. B to A-); if your participation is below average, your grade will be moved down a level or two.
• Writing
• You will submit roughly 5 written solutions over the course of the quarter. The questions will be taken from those discussed in class. Therefore, the emphasis will be not on correctness (as correct and complete solutions should be obtained during class), but on style.
• Evaluation: It is my goal to give each student feedback that will improve the quality of their formal, written arguments over the course of the term. There is no final exam, but the final written assignment may be submitted during week 10 or finals week. These written assignments will account for 100% of your final grade, with adjustments made according to your participation as described above. Your goal should be to improve over the course of the term and this grade will reflect that. It is possible for everyone to improve the quality of their written mathematical arguments (even myself).

Please refer to the classroom policies? for details on class attendance and grading. Exceptions to these policies in place for this course:

• each assignment is mandatory (the lowest grade will not be dropped)
• written solutions must be prepared individually
• late assignments will be accepted within reason, habitual tardiness will cause the revocation of this exception

Last year the course average was A-.

My office hours are anytime that I am in my office (KEC 3071) when I am not busy. My busy times are in my calendar. I am generally in my office during regular business hours.