CS6515 EXAM 3 | QUESTIONS AND ANSWERS | 2026 UPDATE | VERIFIED | 100%
CORRECT
The Class P - (ANSWER)A solution may be found in polynomial time
The Class NP - (ANSWER)A solution may be verified in polynomial time
NP-Hard - (ANSWER)We are yet to find a polynomial time solution
NP-Complete - (ANSWER)Both in NP and at least as hard as NP-Hard problems
NP Reduction - Steps - (ANSWER)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 - (ANSWER)Feasible region is convex
LP: Optimum achieved at a vertex except: - (ANSWER)1. Infeasible - feasible region is empty
2. Unbounded - optimal value of objective function is arbitrarily large
, CS6515 EXAM 3 | QUESTIONS AND ANSWERS | 2026 UPDATE | VERIFIED | 100%
CORRECT
Independent Set -> Vertex Cover - (ANSWER)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 - (ANSWER)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 - (ANSWER)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 - (ANSWER)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 - (ANSWER)Input: C is a CNF whose clauses have at most 3 literals
Output: Assignment of each variable s.t. the CNF is True
Clique - (ANSWER)Input: G is an undirected, unweighted graph, k is the size of the clique
Output: A Clique in G with at least k vertices
CORRECT
The Class P - (ANSWER)A solution may be found in polynomial time
The Class NP - (ANSWER)A solution may be verified in polynomial time
NP-Hard - (ANSWER)We are yet to find a polynomial time solution
NP-Complete - (ANSWER)Both in NP and at least as hard as NP-Hard problems
NP Reduction - Steps - (ANSWER)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 - (ANSWER)Feasible region is convex
LP: Optimum achieved at a vertex except: - (ANSWER)1. Infeasible - feasible region is empty
2. Unbounded - optimal value of objective function is arbitrarily large
, CS6515 EXAM 3 | QUESTIONS AND ANSWERS | 2026 UPDATE | VERIFIED | 100%
CORRECT
Independent Set -> Vertex Cover - (ANSWER)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 - (ANSWER)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 - (ANSWER)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 - (ANSWER)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 - (ANSWER)Input: C is a CNF whose clauses have at most 3 literals
Output: Assignment of each variable s.t. the CNF is True
Clique - (ANSWER)Input: G is an undirected, unweighted graph, k is the size of the clique
Output: A Clique in G with at least k vertices
