CS6515 Exam 3 || 100% Verified Answers.
The Class P correct answers A solution may be found in polynomial time
The Class NP correct answers A solution may be verified in polynomial time
NP-Hard correct answers We are yet to find a polynomial time solution
NP-Complete correct answers Both in NP and at least as hard as NP-Hard problems
NP Reduction - Steps correct answers 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 correct answers Feasible region is convex
LP: Optimum achieved at a vertex except: correct answers 1. Infeasible - feasible region is empty
2. Unbounded - optimal value of objective function is arbitrarily large
Independent Set -> Vertex Cover correct answers 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 correct answers 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 correct answers 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 correct answers 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 correct answers Input: C is a CNF whose clauses have at most 3 literals
Output: Assignment of each variable s.t. the CNF is True
Clique correct answers Input: G is an undirected, unweighted graph, k is the size of the clique
Output: A Clique in G with at least k vertices
Independent Set correct answers Input: G is an undirected, unweighted graph, k is the size of the
IS
Output: An IS in G with at least k vertices
VertexCover correct answers Input: G is an undirected, unweighted graph, b is a budget of the
vertex cover
Output: A vertex cover of G with at most b vertices
RudrataPath correct answers Input: G = (V, E) is an undirected, unweighted graph, s is the
starting node and t is the destination node.
Output: A simple path starting at s and ending at t that passes through each vertex in the graph
exactly once
RudrataCycle correct answers Input: G=(V, E) is an undirected, unweighted graph
Output: A cycle that visits each vertex in the graph exactly once
SubsetSum correct answers Input: A is an array of integers and t is an integer
Output: An array of integers that is a subset of A and sums to t
KnapsackSearch correct answers Input: W is an array of weights, V is an array of values, B is the
capacity of the knapsack, and g is the goal.
Output: Array of items of value at least g with weight less than or equal to B
TSP correct answers Input: G is a weighted, fully connected graph with weights for each n(n-1)/2
edges; b is a budget
Output: A path that visits every vertex in the graph exactly once and has a total cost of b or less
The Class P correct answers A solution may be found in polynomial time
The Class NP correct answers A solution may be verified in polynomial time
NP-Hard correct answers We are yet to find a polynomial time solution
NP-Complete correct answers Both in NP and at least as hard as NP-Hard problems
NP Reduction - Steps correct answers 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 correct answers Feasible region is convex
LP: Optimum achieved at a vertex except: correct answers 1. Infeasible - feasible region is empty
2. Unbounded - optimal value of objective function is arbitrarily large
Independent Set -> Vertex Cover correct answers 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 correct answers 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 correct answers 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 correct answers 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 correct answers Input: C is a CNF whose clauses have at most 3 literals
Output: Assignment of each variable s.t. the CNF is True
Clique correct answers Input: G is an undirected, unweighted graph, k is the size of the clique
Output: A Clique in G with at least k vertices
Independent Set correct answers Input: G is an undirected, unweighted graph, k is the size of the
IS
Output: An IS in G with at least k vertices
VertexCover correct answers Input: G is an undirected, unweighted graph, b is a budget of the
vertex cover
Output: A vertex cover of G with at most b vertices
RudrataPath correct answers Input: G = (V, E) is an undirected, unweighted graph, s is the
starting node and t is the destination node.
Output: A simple path starting at s and ending at t that passes through each vertex in the graph
exactly once
RudrataCycle correct answers Input: G=(V, E) is an undirected, unweighted graph
Output: A cycle that visits each vertex in the graph exactly once
SubsetSum correct answers Input: A is an array of integers and t is an integer
Output: An array of integers that is a subset of A and sums to t
KnapsackSearch correct answers Input: W is an array of weights, V is an array of values, B is the
capacity of the knapsack, and g is the goal.
Output: Array of items of value at least g with weight less than or equal to B
TSP correct answers Input: G is a weighted, fully connected graph with weights for each n(n-1)/2
edges; b is a budget
Output: A path that visits every vertex in the graph exactly once and has a total cost of b or less
