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.

中文翻译：

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建立NP问题难解性的“扩展解集基数”标准

任何问题的难解性与其解决方案的随机性有着明显的直观联系。然而，迄今为止的挑战是，没有切实可行的方法来确定问题的解决方案是否实际上是随机的（或者它是否具有一些隐藏的未发现结构，一旦被检测到就会使其变得非随机）。这阻止了硬问题（例如 NP）绝对难以处理的结论性声明。为了解决这个问题，开发了一个称为序列“可扩展性”的概念。基于此，设想了一个称为“扩展解集的基数”的标准来确定任何序列的（非）随机性。此外，这可以用来确定任何问题的（不）可处理性，这取决于其解决方案是随机的还是非随机的。该标准适用于 2-SAT、3-SAT 和近似硬度等问题，以分析其（不）易处理性。最后，还给出了基于相同标准的唯一博弈猜想有效性的证明。