CS6515 Exam 3 Questions with Verified Correct
Answers
The Class P
A solution may be found in polynomial time
The Class NP
A solution may be verified in polynomial time
NP-Hard
We are yet to find a polynomial time solution
NP-Complete
Both in NP and at least as hard as NP-Hard problems
NP Reduction - Steps
1. Demonstrate that problem B is in the class of NP Problems
- validate a solution in polynomial time
2. Demonstrate that problem B is at least as hard as a problem believed to be NP-Complete.
This is done via a reduction from a known problem A (A->B)
a) Show how an instance of A is converted to B in polynomial time
b) Show how a solution to B can be converted to a solution for A, again in polynomial time
c) Show that a solution for B exists IF AND ONLY IF a solution to A exists
- most prove both parts: if you you have a solution to B, you have a solution to A
- If there is no solution to B, then no solution exists to A
LP: Why optimum occurs at a vertex
, Feasible region is convex
LP: Optimum achieved at a vertex except:
1. Infeasible - feasible region is empty
2. Unbounded - optimal value of objective function is arbitrarily large
Independent Set -> Vertex Cover
Lemma: I is an independent set of G(V, E) iff V - I is a vertex cover of G
Simply check if there is a vertex cover of size V - b in G(V, E). If there is, output V - S
3SAT -> Independent Set
Lemma: f is satisfiable iff the resulting set has an independent set of size m (# of clauses in f)
in G(V, E).
To construct G(V, E), create a node for each literal in each clause and connect them by an
edge. Also connect any literal with it's negation.
Independent Set -> Clique
Lemma: G(V, E) has an independent set of size g iff G'(V, E) has a clique of size g.
To construct G'(V, E), add all the vertices in G(V, E) to G'(V, E) and add edges to G'(V, E) if
there is no edge in G(V, E)
SAT
Input: C is a CNF with any # of variables (n) and clauses (m)
Output: assignment of each variable s.t. the CNF is True
3SAT
Answers
The Class P
A solution may be found in polynomial time
The Class NP
A solution may be verified in polynomial time
NP-Hard
We are yet to find a polynomial time solution
NP-Complete
Both in NP and at least as hard as NP-Hard problems
NP Reduction - Steps
1. Demonstrate that problem B is in the class of NP Problems
- validate a solution in polynomial time
2. Demonstrate that problem B is at least as hard as a problem believed to be NP-Complete.
This is done via a reduction from a known problem A (A->B)
a) Show how an instance of A is converted to B in polynomial time
b) Show how a solution to B can be converted to a solution for A, again in polynomial time
c) Show that a solution for B exists IF AND ONLY IF a solution to A exists
- most prove both parts: if you you have a solution to B, you have a solution to A
- If there is no solution to B, then no solution exists to A
LP: Why optimum occurs at a vertex
, Feasible region is convex
LP: Optimum achieved at a vertex except:
1. Infeasible - feasible region is empty
2. Unbounded - optimal value of objective function is arbitrarily large
Independent Set -> Vertex Cover
Lemma: I is an independent set of G(V, E) iff V - I is a vertex cover of G
Simply check if there is a vertex cover of size V - b in G(V, E). If there is, output V - S
3SAT -> Independent Set
Lemma: f is satisfiable iff the resulting set has an independent set of size m (# of clauses in f)
in G(V, E).
To construct G(V, E), create a node for each literal in each clause and connect them by an
edge. Also connect any literal with it's negation.
Independent Set -> Clique
Lemma: G(V, E) has an independent set of size g iff G'(V, E) has a clique of size g.
To construct G'(V, E), add all the vertices in G(V, E) to G'(V, E) and add edges to G'(V, E) if
there is no edge in G(V, E)
SAT
Input: C is a CNF with any # of variables (n) and clauses (m)
Output: assignment of each variable s.t. the CNF is True
3SAT
