CS6515 EXAM 3 QUESTIONS AND
ANSWERS GRADED A+ 2026
The Class P - ANS A solution may be found in polynomial time
The Class NP - ANS A solution may be verified in polynomial time
NP-Hard - ANS We are yet to find a polynomial time solution
NP-Complete - ANS Both in NP and at least as hard as NP-Hard problems
NP Reduction - Steps - ANS 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
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, 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 - ANS Feasible region is convex
LP: Optimum achieved at a vertex except: - ANS 1. Infeasible - feasible region is empty
2. Unbounded - optimal value of objective function is arbitrarily large
Independent Set -> Vertex Cover - ANS 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 - ANS 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 - ANS 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 - ANS Input: C is a CNF with any # of variables (n) and clauses (m)
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ANSWERS GRADED A+ 2026
The Class P - ANS A solution may be found in polynomial time
The Class NP - ANS A solution may be verified in polynomial time
NP-Hard - ANS We are yet to find a polynomial time solution
NP-Complete - ANS Both in NP and at least as hard as NP-Hard problems
NP Reduction - Steps - ANS 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
@COPYRIGHT 2026/2027 ALLRIGHTS RESERVED 1
, 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 - ANS Feasible region is convex
LP: Optimum achieved at a vertex except: - ANS 1. Infeasible - feasible region is empty
2. Unbounded - optimal value of objective function is arbitrarily large
Independent Set -> Vertex Cover - ANS 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 - ANS 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 - ANS 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 - ANS Input: C is a CNF with any # of variables (n) and clauses (m)
@COPYRIGHT 2026/2027 ALLRIGHTS RESERVED 2
