{
"translations": {
"en": {
"direction": {
"type": "multiple",
"description": "Don't forget: A proof to show that problem X is NP-hard involves reducing known NP-hard problem H to X. When doing NP-completeness proofs, it is very important not to get this reduction backwards! If you get it backwards, all that you have done is come up with an expensive algorithm for X!
However, when we reduce known hard problem H to candidate problem X, that means we are using X as a step to solve H. And if we know that H is hard, that means X must also be hard (because if X were not hard, then neither would H be hard).",
"question": "Reduce problem A to B means that we use A as a step to solve B",
"answer": "False",
"choices": ["True", "False"],
"correctFeedback": ["Reducing problem A to B means that We use B as a step to solve A. In other words, B is the algorithm in the black box in the reduction template."]
},
"hard": {
"type": "select",
"description": "Don't forget: A proof to show that problem X is NP-hard involves reducing known NP-hard problem H to X. When doing NP-completeness proofs, it is very important not to get this reduction backwards! If you get it backwards, all that you have done is come up with an expensive algorithm for X!
However, when we reduce known hard problem H to candidate problem X, that means we are using X as a step to solve H. And if we know that H is hard, that means X must also be hard (because if X were not hard, then neither would H be hard).",
"question":"If we reduce A to B, which of the following statements are true?",
"answer": ["If A is hard, then B must be hard"],
"choices": ["If A is hard, then B must be hard", "If A is not hard, then B must not be hard", "None of the above"]
},
"decision": {
"type": "multiple",
"description": "Cook proved that SAT is NP-hard. Explaining Cook's proof is beyond the scope of this course. But we can briefly summarize it as follows. Any decision problem F can be recast as some language acceptance problem L:
F(I) = YES ⇔ L(I') = ACCEPT.
That is, if a decision problem F yields YES on input I, then there is a language L containing string I′ where I′ is some suitable transformation of input I. Conversely, if F would give answer NO for input I, then I 's transformed version I′ is not in the language L.",
"question": "There is a language L containing string I' if a decision problem F yields YES on input I",
"answer": "True",
"choices": ["True", "False"]
},
"TM": {
"type": "select",
"description":"Turing machines are a simple model of computation for writing programs that are language acceptors. There is a universal
Turing machine that can take input as a description for a Turing machine, and an input string, and return the execution of that machine on that string.
This Turing machine in turn can be cast as a Boolean expression such that the expression is satisfiable if and only if the Turing machine yields ACCEPT for that string.",
"question":"which of the following statements are true?",
"answer": ["Turing machines are a simple model of computation for writing programs that are language acceptors", "Universal
Turing machine can take input as a description for a Turing machine and an input string, and return the execution of that machine on that string", "Turing machine can be cast as a Boolean expression such that the expression is satisfiable if and only if the Turing machine yields ACCEPT for that string"],
"choices": ["Turing machines are a simple model of computation for writing programs that are language acceptors", "Universal
Turing machine can take input as a description for a Turing machine and an input string, and return the execution of that machine on that string", "Turing machine can be cast as a Boolean expression such that the expression is satisfiable if and only if the Turing machine yields ACCEPT for that string"]
},
"q7": {
"type": "multiple",
"description": "To show that a decision problem X is NP-complete, we prove that X is in NP (normally easy, and normally done by giving a suitable polynomial-time, non-deterministic algorithm) and then prove that X is NP-hard.
To prove that X is NP-hard, we choose a known NP-complete problem, say A. We describe a polynomial-time transformation that takes an arbitrary instance I of A to an instance I′ of X. We then describe a polynomial-time transformation from SLN′ to SLN such that SLN is the solution for I.",
"question": "To show a decision problem A is NP-complete, we need to prove A is in NP and then A is in NP-hard",
"answer": "True",
"choices": ["True", "False"]
}
}
}
}