Outline of Material for Test #3, CS8813, Fall 2004

Intractable Problems

• Definitions and Background
• Be able to define P and NP classes of problems
• How do we determine how much time it takes for a non-deterministic Turing machine to solve a problem?
• What is a polynomial-time reduction and what is its utility?
• What does it mean to say a problem in NP-complete? How about NP-hard?
• What is the satisfiability problem?
• What is the definition of the conjunctive normal form (CNF)?
• What is a the 3SAT problem?
• What is the Problem of Independent Sets (IS)?
Remember the difference between a problem and an algorithm.
• What is the Node-Cover Problem?
• What is a Hamilton-Circuit?
• What is the directed Hamilton-Circuit problem?
• What is the undirected Hamilton-Circuit problem?
• Constructions
• SAT NP-complete proof constructions
• Be able to describe the role and purpose of the terms Em,w = S AND N AND F AND U (Page 434). What do each of the terms do? (S, N, F, and U) (Recall the errata as the book has a mistake and doesn't include U, the test won't make that mistake....)
• Be able to describe how we construct the Ni term, in particular what the Aij and Bij terms do and how they are formed.
• Be able to discuss how the construction of Em,w can be performed in polynomial time
• NP-completeness of CSAT
• Be able to convert an arbitrary expression into Conjunctive Normal Form (CNF) using the construction given on pages 437-440. (Also see homework).
• When doing this construction, be sure to do it in steps so that I can give partial credit if a simple mistake is made at an early step.
• NP-completeness of 3SAT
• Be able to convert an arbitrary CSAT expression into 3SAT form using the construction on page 445.
• Problem of Independent Sets
• Be able to show how to convert a 3SAT expression into an equivalent IS problem.
• Be able to argue that the IS reduction can be performed in polynomial time.
• Directed Hamilton-Circuit Problem (DHC)
• Be able to do the construction of the DHC problem on a simple 3SAT expression (such as figure 10.10).
• Proofs
• Be able to prove that for every boolean expression, there exists an equivalent expression where negations only occur in literals
See page 439
• Be able to prove that NC is NP-complete through polynomial-time reduction from IS.
• Be able to prove that HC is NP-complete through polynomial-time reduction from DHC
• Be able to prove that TSP is NP-complete through polynomial-time reduction from HC.
• Review All Homework!