Cryptocurrency Q&A
Does apx hard imply NP hard?
Does apx hard imply NP hard?
TaegeukChampionship
Sat Sep 14 2024
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6 answers
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Could you please clarify for me the relationship between "APX-hard" and "NP-hard" in the context of computational complexity? Specifically, I'm wondering if a problem being APX-hard necessarily implies that it is also NP-hard. Could you elaborate on the implications and potential differences between these two classifications, and provide any examples to further illustrate your points?
6 answers
Giulia
Mon Sep 16 2024
The relationship between NP and APX is intricate yet intriguing. On a conceptual level, one might be tempted to equate the two, but a closer inspection reveals their fundamental differences.
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MysticEchoFirefly
Mon Sep 16 2024
NP comprises a set of decision problems, characterized by the existence of polynomial-time verifiers for solutions. Conversely, APX encapsulates optimization problems, which aim to find the best solution among a set of feasible alternatives.
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CryptoQueen
Sun Sep 15 2024
Despite their apparent dissimilarities, NP and APX share a subtle interplay. Conceptually, one can envision a bidirectional Flow between these two classes, allowing for insights and techniques to be transferred from one domain to the other.
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TaegeukChampion
Sun Sep 15 2024
This connection becomes evident when considering NP-complete decision problems. These notoriously difficult problems, which defy efficient solutions, often have corresponding optimization versions.
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Moonshadow
Sun Sep 15 2024
These optimization counterparts, however, pose an even greater challenge, falling into the APX-hard category. This designation signifies that not only are they inherently complex, but they also resist approximation algorithms that guarantee near-optimal solutions within a reasonable time frame.
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