确定NP问题难解性的“扩展解集基数”准则
The "cardinality of extended solution set" criterion for establishing the intractability of NP problems
摘要：
任何问题的棘手性和其解决方案的随机性都有明显的直观联系。然而，到目前为止的挑战是，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（或者它是否有一些隐藏的未被发现的结构，一旦被发现就会使其成为非随机的）。这就防止了硬问题（如NP）被断定为绝对难以处理。为了处理这个问题，一个被称为“可扩展性”的序列的概念。在此基础上，提出了一个判断序列随机性的标准，称为“扩展解集基数”。此外，这可以用来建立任何问题的（不）易处理性，这取决于它的解决方案是随机的还是非随机的。将该准则应用于2-SAT、3-SAT和逼近困难等问题，分析了它们的易处理性。最后，给出了基于同一准则的唯一对策猜想的有效性证明。
英文摘要：
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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