The "cardinality of extended solution set" criterion for establishing the intractability of NP problems
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The intractability of any problem and the randomness of its solutions have an obvious intuitive connection. However, the challenge till now has been that there is no practical way to firmly establish if the solution to a problem is actually random (or whether it has some hidden undiscovered structure, which upon being detected would render it non-random). This has prevented the conclusive declaration of hard problems (such as NP) as being definitely intractable. For dealing with this, a concept called "extensibility" of a sequence is developed. Based on this, a criterion termed as "cardinality of extended solution set" is conceived to ascertain the (non)randomness of any sequence. Further, this can then be used to establish the (in)tractability of any problem depending on whether its solutions are random or non-random. This criterion is applied to problems such as 2-SAT, 3-SAT and hardness of approximation to analyze their (in)tractability. Finally, a proof for the validity of the Unique Games Conjecture based on the same criterion is also presented.
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