Lecture Summaries for 308-360B
also check: announcements
assignments return to: course home page
You will need the lp package lp_solve for the next
part of the course, please check announcements for details.
Notes for Linear Programming:
Either: fukuda.pdf or fukuda.ps
Notes for NP-completenss: Handout given in
class, also available at Copi-EUS.
For office hours during exam period, please see
April 10: Final monologue and question period. Good luck on the exam.
April 8: Reductions from SAT to 3-SAT, 3-SAT to Subset Sum. Read: Sections
April 3: More reductions. Ind. set to clique and
vertex cover, Sat to 3-Sat. Finding the "boundary" between P and
NP-complete. Read: Text Ch 34, 966-986
April 1: NP-hard vs NP-complete. Cook's theorem
and it implications. How to prove problems are NP-complete using polynomial
time reductions. First reduction: sat to ind. set. Assigment
4 is posted.
Mar 27: Definitions of polynomial reducibility, non deterministic algorithms
and the class NP.
Emphasis on checking the "yes" answer in polynomial time by use of a certificate.
Read: Read NP-completeness handout
Mar 25: Introduction to NP completeness. Easy and
hard problems. Examples of obtaining hard problems from easy ones by
small modifications. Properties of "hard" problems.
Mar 20: Bohdan's lecture on finding a feasible solution to a system of
inequalities using the b-rule.
Read: brule.ps (brule.pdf) at least first 4 pages.
Mar 18: Certificates of optimality, infeasibility
and unboundedness. The weak duality theorem.
Read: LP notes up to and including Section 2.4.
Mar 13: Introduction to linear programming,
the wine production problem. Formulation of a LP in standard form, and integer
linear program (ILP). Introduction to proof of optimality via positive combinations
Read: LP notes up to P. I-5.
Mar 11: Max flow min cut theorem, Integrality theorem.
We used this theorem to prove that a bipartite graph in which each vertex
has degree k has a perfect matching. Variants to guarantee polynomial time:
augmenting path using minimum number of edges (Edmonds-Karp) and augmenting
path of maximum capacity.
March 8: Discussion of midterm. Implementation of
the network flow algorithm.
Mar 4: Midterm
Feb 20: Augmenting Paths. Matchings. Relationship of matching algorithm
using alternating paths to network flow algorithm using augmenting paths.
Read: handout from Feb 11 up to, but not including, the Augmenting
Feb 18: Network Flows. Basic concepts, augmenting
Feb 13: The Stable Marriage Problem. Today
we learned the important:
Lesson in love: Unless you are fantastically good-looking, you
need to take the initiative.
Read: David Feldman's notes,
and the New Scientist article,
which contains the above quote.
Feb 11: Bipartite matchings. Solution by alternating paths. A
Vertex Cover for a graph G is a subset S of
vertices such that each edge of G has at least one of its endpoints
in S. We saw that for any matching M and any vertex cover S, the number
of edges in M is at most the number of vertices in S.
Get handout Matchings and Flows from
Copi-EUS, ground floor of McConnell Engineering. Read section 7.10 of
Feb 6: Johnson's algorithm. Read: Section 25.3
Feb 4: Disjkstra's algorithm for directed graphs with non-negative
edge weights. Comparison with Prim's algorithm. Checking the correctness
of a shortest paths treee by one iteration of the Bellman-Ford algorithm.
Read: Chapter 24 up to the end of Section 24.3
Jan 30: Introduction to directed graphs and shortest path problems. Bellman-Ford
algorithm for directed graphs with negative edge weights.
Read: Bruce Reed's handout(pdf) and Section 24.1.
Jan 28: More examples of dynamic programming. Paths through a grid. Knapsack
Problem (see Ex. 16.2-2, p. 384). C code for knapsack problem and sample
run is here.
Read: David Bryant's Handout(pdf) .
Jan 23: Introduction to dynamic programming. Matrix chain multiplication.
Read: David Bryant's Handout(pdf). Section 15.2.
Jan 21: The algorithms of Prim and Kruskal, and implementations in O(mlogn)
Read: Finish reading chapter
23. Tutorials start today!
Jan 16: Proof of the optimality of the greedy algorithm (I added this
proof to the course notes selection.(pdf)) Introduction to minimum spanning
trees. Kruskal's algorithm viewed as an example of greedy selection.
Read: Section 16.4, Start reading
Jan 14: Introduction to selection problems, weighted selection problems
and the greedy algorithm.
Read: Course notes on selection (pdf) and do the exercises!
Jan 9: Outline of material to be covered in the course. General
notions of easy and hard problems.
Greedy algorithms. Sushi (knapsack) problem.
Read: Sections 16.1, 16.2
( 17.1, 17.2 first edition).
Exercise: Prove the greedy heuristic finds an optimal
solution for the fractional knapsack problem.
Jan 7: Introduction to course via statement of two problems: the assignment
problem and the travelling salesman problem.