CS 787 
Course Log

Lecture 1
Friday Sept 3

  Ways to compute the greatest common divisor: exhaustive
  search, factoring, subtractive and division versions of
  Euclid's algorithm.  Lame's theorem (logarithmic number of
  division steps).

Lecture 2
Wednesday Sept 8

  Induction as an algorithm design tool

  Manber, Using induction to design algorithms, CACM 1988
  Examples:  Gaussian elimination
             Largest permuted set

Lecture 3
Friday Sept 10

  Dynamic Programming
  Example: Longest increasing subsequence

  Pruhs, How to design dynamic programming algorithms sans recursion,
  SIGACT News 1988

Lecture 4
Monday Sept 13

  Divide and Conquer

  Cauchy's proof that geometric mean <= arithmetic mean
  Karatsuba subquadratic algorithm for integer multiplication

  Cauchy's proof is in Polya and Szego, Problems and Theorems in
                                        Analysis, vol. 1, p. 64.
  Karatsuba and Ofman, Soviet Physics Doklady, 1963.

Lecture 5
Wednesday Sept 15

  Intro to linear programming 

  Lectures by Richard Karp [Great Algorithms -- CS 292-5, 2006].

Lecture 6
Friday Sept 17

  The simplex algorithm (matrix form)

  Karp, op. cit.

Lecture 7
Monday Sept 20

  Duality theorem (strong and weak versions)

  Karp, op. cit.

Lecture 8
Wednesday Sept 22

  The complexity of LP
  Simplex doesn't take shortest paths to an optimal vertex (2-d example)
  Simplex can take exponential time (Klee-Minty)
  Average-case and smoothed analysis, briefly

  Papadimitriou and Steiglitz, op. cit., 8.6
  Spielman and Teng, Proc. 2001 STOC

Lecture 9
Friday Sept 24

  Single-source shortest path as LP
  Dual problem: costs to the sink

  Papadimitriou and Steiglitz, Combinatorial Optimization, Section 3.4

Lecture 10
Monday Sept 27

  Maximum network flow and its dual
  Max flow - min cut theorem
  Ford-Fulkerson augmenting path algorithm

  Lawler, Combinatorial Optimization, Sections 4.1, 4.2, 4.6 

[Note: The next three lectures were by Jin-Yi Cai.]

Lecture 11
Wednesday Sept 29

  Duality as a tool for estimating solution quality

  V. V. Vazirani, Approximation Algorithms, Section 12.1

Lecture 12
Friday Oct 1

  Flows and cuts, unimodularity of network matrices

  Vazirani, op. cit., 12.2

Lecture 13
Monday Oct 4

  A dual-fitting approximation algorithm for set cover

  V. Vazirani, op. cit., 13.1

Lecture 14
Wednesday Oct 6

  Two network flow applications:

  Bipartite matching
  Baseball elimination

  K. Wayne, Proc. 2010 SODA

Lecture 15
Friday Oct 8

  Baseball elimination (continued)

No class
Monday Oct 11

Lecture 16
Wednesday Oct 13

  Min-cost flow problems
  Examples: shortest path
            job assignment problem

  Hungarian algorithm for job assignment (proof of termination)

  E. Bach, The Hungarian Algorithm Sans Dual Variables, notes.
  Papadimitriou & Steiglitz, Section 11.2

Lecture 17
Friday Oct 15

  Matchings and vertex covers
  Koenig-Egervary theorem
  Strongly polynomial bound for Hungarian algorithm ( O(n^4) )

Lecture 18
Monday Oct 18

  O(n^3) Hungarian implementation, using better data structures
  Linear Programs with Greedy Solutions (Greed before Edmonds)

  A. Hoffman, Greedy Algorithms that Succeed

Lecture 19
Wednesday Oct 20

  Transportation problems
  Monge inequalities
  Interval graph coloring

Lecture 20
Friday Oct 22

  Boruvka/Kruskal minimum spanning tree algorithm
  Introduction to matroids

  J. Edmonds, Matroids and the Greedy Algorithm, 1971

Lecture 21
Monday Oct 25

  Correctness of GREEDY for matroids
  Partition matroids
  "Polygamous" bipartite matchings

Lecture 22
Wednesday Oct 27

  Transversal matroids
  Unit-job scheduling with deadlines and penalties

Lecture 23
Friday Oct 29

  Matroids and linear programming
  Matroid polyhedra
  Max-weight independent set and its dual (GREEDY solves both)

Lecture 24
Monday Nov 1

  New topic: Randomized Algorithms

  Array search under a "density promise" (50% 0's, 50% 1's).
  Introduction to finite fields

Lecture 25
Wednesday Nov 3

  Polynomials in one variable
  Cipolla's algorithm for square roots mod p

  Bach & Shallit, Algorithmic Number Theory I, pp. 157-158.

Lecture 26
Friday Nov 5

  Multivariate polynomials
  Randomized identity testing

  J. Schwartz, J. ACM 1980

Lecture 27
Monday Nov 8

  Matching in general graphs
  Tutte polynomial
  Reducing matching to zero testing

Lecture 28
Wednesday Nov 10

  MVV algorithm: maximum matching from one matrix inversion

  Mulmuley, Vazirani, Vazirani, Combinatorica v. 7, 1987, 105-113

Lecture 29
Friday Nov 12

  Combinatorial algorithm for matching

  Edmonds, Canadian J. Math, 1965
  Chapter 9 of Tarjan, Data Structures and Network Algorithms (SIAM 1983)

Lecture 30
Monday Nov 15

  Matching polytope
  General graph extension of Koenig-Egervary (Edmonds)

  Edmonds, loc. cit.
  
Lecture 31
Wednesday Nov 17

  Randomized rounding
  Vertex cover (briefly)
  Wire routing

  Raghavan & Thompson, Combinatorica 7:4, 1987, 365-374.

Lecture 32
Friday Nov 19

  Randomized rounding for MAX-SAT (unit weights)

  Goemans & Williamson, SIAM J. Discrete Math., 1994.

Lecture 33
Friday Nov 19

  Convex sets and functions
  Introduction to convex optimization

  S. Boyd and L. Vandenberghe, Lecture notes for EE 392X, 
  Stanford University.

Lecture 34
Monday Nov 22

  Positive definite matrices
  Semidefinite programming

Lecture 35
Wednesday Nov 24

  Approximating MAX-CUT via semidefinite programming

  Goemans & Williamson, J. ACM 1995

Lecture 36
Monday Nov 29

  MAX-CUT (cont'd)
  Algorithms for MIN-CUT: network flow, edge shrinking

  Motwani & Raghavan, pp. 7-9

Lecture 37
Wednesday Dec 1

  Introduction to the Ellipsoid algorithm

  Aspvall & Stone, J. Algorithms, v. 1, 1980.
  Lawler, The Great Mathematical Sputnik of 1979.

Lecture 37
Friday Dec 3

  A polynomial-time algorithm for linear programming
  Aspvall & Stone, op. cit.

  See also: Papadimitriou & Steiglitz, pp. 170-184.

Lecture 38
Monday Dec 6

  Polarity (a variant of classical geometric duality)
  Separation oracles from optimization algorithms
  Matroid intersection examples: bipartite matching, directed spanning trees

Lecture 39
Wednesday Dec 8

  Integrality for matroid polyhedral intersection
  Ellipsoid-based poly time algorithm for matroid intersection problems

Lecture 40
Friday Dec 10

  Odds and ends:
  Combinatorial separation oracles (e.g. for directed spanning trees)
  Separation oracle for trace-zero semidefinite matrices

Lecture 41
Monday Dec 13

  Introduction to submodular functions
  Examples: cuts in graphs, matroid rank, cooperative games

  Course evaluation

Lecture 42
Wednesday Dec 15

  Survey of algorithmic problems for submodular functions and polymatroids
  Randomized approximations for submodular maximization 

  Feige, Mirrokni, and Vondrak, Maximizing non-monotone submodular
  functions, manuscript, 2009.  (Preliminary version in FOCS 2007.)