Midterm Review

Network flow (~ 45%)

• Use algorithms: Examples
  1. Given a flow network, use the Edmonds-Karp algorithm to determine the maximum flow, and a cut corresponding to the maximum flow.

• Theoretical questions: Examples
  1. Prove properties of network flows, such as CLR: exercise 26.1-1 (if (u,v) and (v,u) are not in E, then f(u,v) = f(v,u) = 0).
  2. Prove relationships between flows, networks, and residual networks, such as lemma 26.2.
  3. Prove properties of cuts and relationships between cuts and flows, such as the max-flow min-cut theorem.
  4. Give counter-examples to disprove statements. Example: Let F = maxu C({u}, V). Any flow is > F.
  5. Derive the time complexity of the Edmonds-Karp algorithm.

Approximation algorithms (~ 55%)

• Use algorithms: Examples
  1. Factor 2 greedy algorithm for cardinality vertex cover.
  2. Factor Hn greedy algorithm for set cover.
  3. FPTAS for Knapsack.
  4. Given a linear program, write it in standard form.
  5. Given a linear program, write its dual.
  6. Given a set cover or vertex cover instance, formulate it as an integer linear program and give its relaxation. Given a solution to the relaxed version, show the set cover that results from applying the rounding algorithm.
  7. Primal-dual algorithm for set cover.
  8. Given an instance of MAX-SAT, show the result from the derandomized factor 1/2 algorithm.

• Theoretical questions: Examples
  1. Prove approximation factors or the correctness of algorithms that we have discussed.
  2. Prove approximation factors or correctness of new algorithms that are presented, such as problem 3 in HW 2 (HW2 does not ask you to prove these, but you should be able to).
  3. Give tight examples for new algorithms that you are given, such as problem 3 in HW 2.
  4. Give an algorithm having a specified approximation factor, such as the factor 1/2 approximation algorithm for the acyclic subgraph problem.
  5. Prove any properties that you are asked to prove!
  6. Disprove statements, using counter-examples.
  7. Formulate a specified problem as an integer linear program, such as the Knapsack problem.
  8. Prove properties of primal and dual, such as the weak duality theorem and complementary slackness conditions.
  9. Given a randomized algorithm, derandomize it to yield a deterministic algorithm that performs at least as well as the expected behavior of the randomized one.

Notes:

1. You should show steps in questions that ask you to use an algorithm.
2. You should justify answers to theoretical questions.
3. You may use facts that we have proved in class, such as the max-flow min-cut theorem, unless the question forbids it.
4. The exam is closed book and closed notes.

Weights

• Use algorithms: ~ 50 %
• Theoretical questions: ~ 50 %