Heuristic algorithms have been used for a long time to tackle problems that are known or conjectured intractable. A heuristic algorithm is one that provides a correct decision for most inputs, but may fail on some. We focus on the case when failure means that the algorithm does not return any answer, rather than returning a wrong result. These algorithms are called errorless heuristics.

A reasonable quality measure for heuristics is the failure rate over the set of n-bit instances. When no efficient exact algorithm is available for a problem, then, ideally, we would like one with vanishing failure rate. We show, however, that this is hard to achieve: unless a complexity theoretic hypothesis fails, some NP-complete problems cannot have a polynomial-time errorless heuristic algorithm with any vanishing failure rate.

On the other hand, we prove that vanishing, even exponentially small, failure rate is achievable, if we use a somewhat different accounting scheme to count the failures. This is based on special sets, that we call α-spheres, which are the images of the n-bit strings under a bijective, polynomial-time computable and polynomial-time invertible encoding function α. Our main result is that polynomial-time errorless heuristic algorithms exist, with exponentially low failure rates on the α-spheres, for a large class of decision problems. This class includes, surprisingly, all known intuitively natural NP-complete problems. We also explore some connections to the theory of average case complexity.

启发式算法已经使用了很长时间，以解决已知的或难以解决的问题。启发式算法是一种可以为大多数输入提供正确决策的算法，但在某些算法上可能会失败。我们关注的情况是失败意味着算法不返回任何答案，而不是返回错误的结果。这些算法称为无错误启发式算法。

启发式的合理质量度量是n位实例集的故障率。当没有有效的精确算法可用于问题时，理想情况下，我们希望故障率消失的算法。但是，我们证明这是很难实现的：除非复杂性理论假设失败，否则某些NP完全问题就不可能拥有多项式时间无差错启发式算法，且失效率几乎没有。

另一方面，我们证明，如果我们使用稍微不同的计费方案来计算故障，则故障率甚至可以消失，甚至成倍减小。这是基于特殊集合的，我们称为α-球体，它们是在双射，多项式时间可计算和多项式时间可逆编码函数α下的n位字符串的图像。我们的主要结果是，存在针对多种决策问题的多项式时间无错误启发式算法，α球上的失效率极低。令人惊讶的是，此类包括所有已知的直观自然NP完全问题。我们还探讨了与平均案例复杂度理论的一些联系。