Review Topics
Chapter Seven
- Know all definitions & concepts associated with flow networks
- Know how to perform the Ford-Fulkerson algorithm (FFA)
- Know how to use the FFA to calculate max flows and min cuts
- Know how to construct residual graphs
- Know how to calculate capacities of cuts and the value of a network
flow based on any cut.
- Know the version of the FFA that runs in O(mC) steps, what that means,
how to explain it, and how to show it is true. Know what m and C
represent in the formula O(mC)
- Know the relationship between flows and cuts.
- Know the second version of the FFA that "chooses good augmenting paths,"
and know the bound on the number of steps the algorithm requires - the
bound that was proved in the text: O(m2logC).
- Know what m and C represent in the bound O(m2logC).
- Know how to explain the Δ parameter of the second version of the
FFA and how it us used to achieve good augmenting paths.
- Understand that the facts we proved about the FFA assume that the flow
networks under consideration have positive integer capacities on all
edges.
- Know how to solve bipartite matching problems using the FFA.
- Know that the following problems can be solved using the FFA:
Survey Design, Airline Scheduling, Image Segmentation, and Baseball
Elimination.
- Know details of how to reduce Survey Design, Image Segmentation and
Baseball Elimination into flow problems that can be solved with the FFA.
For example, know how to take the problem of determining whether the
Giants can win the pennant and transform it into a network flow problem,
run the FFA on the flow problem, and use the results to decide whether
the Giants can win the pennant.
Chapter Eight
- Know the definitions of P, NP and NP-complete.
- Know what a decision problem is.
- Know what an instance of a decision problem is.
- Know about measures of size of an instance of a problem.
- Know the terminology regarding decision problems.
- Know what X≤PY and X≡PY mean for decision
problems X and Y.
- Know about the transitivity of the ≤P relationship - i.e.
X≤PY and Y≤PZ imply X≤PZ.
- Know and understand that if X≤PY and Y∈P,
then X∈P
- Know and understand that if X≤PY and X∉P,
then Y∉P
- Know how the reductions we discussed were done - know the 'constructions'
- Given a description of a problem, be able to say whether it is a problem
our text showed to be in P, in NP, or NP-complete.
- Know techniques for demonstrating that a problem is in P, is in NP, or is
NP-complete.
- Know facts about the question of whether P = NP. What is the
significance of the question? Do we know whether P ⊆ NP?
Do we know whether NP ⊆ P?
- Know what NP stands for (Non-deterministically Polynomial).
- Know the relationship of NP-complete problems to the question of
whether P = NP.