GA exam 3 CS 6515 || 100% Actual Answers.
Search problem correct answers A problem is a search problem if, given an and a potential
solution, you can verify the solution is correct in polynomial time. The em must have the form
where you output a valid solution if one exist, and report no otherwise
Decision problem correct answers Desicion problems are a simplified version of search problem.
They are a the bead rock of NP-C theory because their simple "yes/no" format makes them easy
to use in reductions.
Optimization problems correct answers Optimization problems take the challenge one step
further. They don't just ask for any solution, they ask for the best one. An optimization problem
seeks to find a solution that maximizes or minimizes som value
Polynomial time correct answers Considered efficient or inctractable. Te runtime is bounded by a
polynomial function of the input size: O(n), O(n^2) even O(n^3000)
Exponential time correct answers Considered inefficient or intractable for large inputs. the
runtime grows exponentially with respect to the input size: O(2^n)
Pseudo Polinomial correct answers A special case that seems efficient but it is not. the runtime is
polynomial in the numeric value of the input, but exponential in the size(number of bits)of the
input
Class P correct answers P (polynomial time) this class contains all decision problems that can be
solved by deterministic algorithm in polynomial time
Class NP correct answers NP (Non-deterministic polynomial time) This class contains all
decisions problems where if the answer is "yes " there is a proof at can be verified in polynomial
time
Np-Hard correct answers These re the problems that are at least as hard as the hardest problem in
NP. A problem is NP-Hard if any problem in NP can be reduced to in in polynomial time
NP-Complete correct answers This class represents the hardest NP problems. NPC problems are
equivalent in difficulty. If we find a polynomial time algorithm for any of the NPC problems. we
have found one for all of them. (P=NP)
What make a problem NPC correct answers 1. It is in NP
2. It is in NP-Hard
What are reductions used for? correct answers A Reduction is a formal way to say that Problem
A is no harder than problem B. We denote it as A <=p B. It's a polynomial-time algorithm that
transform any instance of problem A into an equivalent instances of problem B
What is the direction of the reductions: correct answers Kwon to unkown.
, The reduction flow from easier to harder problems in a hierarchical sense. If we say problem A
reduces to problem B, it means that A is no harder than B.
True or False:
Any problem in problem in NP reduces to any problem in NPC correct answers True.
This is the part of the definition of NPC. It's why NPC problems are considered the hardest in NP
True or False:
Any problem in NPC reduces to any problem in NP-Hard correct answers True.
By definition, NP_Hard problems are at least as hard as everything in NP
What does it mean that a problem is known to be in NP correct answers These are problems that
are confirmed to have polynomial time verifies. This include problems in P and all NPC
What does it means that problem is NOT known to be in NP correct answers These are problems
that cannot be part of NP, usually because they are not decision problems or are proven to be
even harder than NP.
Given a example of a problem that is not in NP correct answers The Halting problem
Steps to prove a problem is NP-Complete correct answers 1 Prove the problem is in NP
1.1 Define the certificate/solution: What information would you need to be convinced the answer
exist.
1.2. Design a verifier: Write an algorithm that takes the problem instance and the solution.
1.3. Prove the verifier in polynomial time
2 Prove the problem is NP-Hard.
2.1 choose a known NPC problem A
2.2 Construct a transformation
2.3 Prove the transformation is polynomial
2.4 Prove correctness
How to choose a problem to use for reductions: correct answers - if the problems involves graph,
nodes and connections, a good start Vertex Cover, Clique, Independent Set, or Rudrata path.
- If the problem involves logic, choices constraints or truth assignment. SAT or 3SAT
- If the problem involves numbers, set, packing, or scheduling, a good choice might be SSS or
knapsack
Define SAT (Inputs, outputs) correct answers Definition: Decided if a boolean formula CNF is
satisfiable.
Inputs: A boolean formula in CNF
Output: Assignment if is possible, otherwise.
Define 3SAT (Inputs, outputs) correct answers A SAT variant that clauses has at most 3 literals
Inputs: A boolean formula in CNF
Output: Assignment if is possible, No otherwise.
Search problem correct answers A problem is a search problem if, given an and a potential
solution, you can verify the solution is correct in polynomial time. The em must have the form
where you output a valid solution if one exist, and report no otherwise
Decision problem correct answers Desicion problems are a simplified version of search problem.
They are a the bead rock of NP-C theory because their simple "yes/no" format makes them easy
to use in reductions.
Optimization problems correct answers Optimization problems take the challenge one step
further. They don't just ask for any solution, they ask for the best one. An optimization problem
seeks to find a solution that maximizes or minimizes som value
Polynomial time correct answers Considered efficient or inctractable. Te runtime is bounded by a
polynomial function of the input size: O(n), O(n^2) even O(n^3000)
Exponential time correct answers Considered inefficient or intractable for large inputs. the
runtime grows exponentially with respect to the input size: O(2^n)
Pseudo Polinomial correct answers A special case that seems efficient but it is not. the runtime is
polynomial in the numeric value of the input, but exponential in the size(number of bits)of the
input
Class P correct answers P (polynomial time) this class contains all decision problems that can be
solved by deterministic algorithm in polynomial time
Class NP correct answers NP (Non-deterministic polynomial time) This class contains all
decisions problems where if the answer is "yes " there is a proof at can be verified in polynomial
time
Np-Hard correct answers These re the problems that are at least as hard as the hardest problem in
NP. A problem is NP-Hard if any problem in NP can be reduced to in in polynomial time
NP-Complete correct answers This class represents the hardest NP problems. NPC problems are
equivalent in difficulty. If we find a polynomial time algorithm for any of the NPC problems. we
have found one for all of them. (P=NP)
What make a problem NPC correct answers 1. It is in NP
2. It is in NP-Hard
What are reductions used for? correct answers A Reduction is a formal way to say that Problem
A is no harder than problem B. We denote it as A <=p B. It's a polynomial-time algorithm that
transform any instance of problem A into an equivalent instances of problem B
What is the direction of the reductions: correct answers Kwon to unkown.
, The reduction flow from easier to harder problems in a hierarchical sense. If we say problem A
reduces to problem B, it means that A is no harder than B.
True or False:
Any problem in problem in NP reduces to any problem in NPC correct answers True.
This is the part of the definition of NPC. It's why NPC problems are considered the hardest in NP
True or False:
Any problem in NPC reduces to any problem in NP-Hard correct answers True.
By definition, NP_Hard problems are at least as hard as everything in NP
What does it mean that a problem is known to be in NP correct answers These are problems that
are confirmed to have polynomial time verifies. This include problems in P and all NPC
What does it means that problem is NOT known to be in NP correct answers These are problems
that cannot be part of NP, usually because they are not decision problems or are proven to be
even harder than NP.
Given a example of a problem that is not in NP correct answers The Halting problem
Steps to prove a problem is NP-Complete correct answers 1 Prove the problem is in NP
1.1 Define the certificate/solution: What information would you need to be convinced the answer
exist.
1.2. Design a verifier: Write an algorithm that takes the problem instance and the solution.
1.3. Prove the verifier in polynomial time
2 Prove the problem is NP-Hard.
2.1 choose a known NPC problem A
2.2 Construct a transformation
2.3 Prove the transformation is polynomial
2.4 Prove correctness
How to choose a problem to use for reductions: correct answers - if the problems involves graph,
nodes and connections, a good start Vertex Cover, Clique, Independent Set, or Rudrata path.
- If the problem involves logic, choices constraints or truth assignment. SAT or 3SAT
- If the problem involves numbers, set, packing, or scheduling, a good choice might be SSS or
knapsack
Define SAT (Inputs, outputs) correct answers Definition: Decided if a boolean formula CNF is
satisfiable.
Inputs: A boolean formula in CNF
Output: Assignment if is possible, otherwise.
Define 3SAT (Inputs, outputs) correct answers A SAT variant that clauses has at most 3 literals
Inputs: A boolean formula in CNF
Output: Assignment if is possible, No otherwise.
