We study the planted clique problem in which a clique of size k is planted in an Erd\H{o}s-R\'enyi graph G(n, 1/2), and one is interested in either detecting or recovering this planted clique. This problem is interesting because it is widely believed to show a statistical-computational gap at clique size k=sqrt{n}, and has emerged as the prototypical problem with such a gap from which average-case hardness of other statistical problems can be deduced. It also displays a tight computational connection between the detection and recovery variants, unlike other problems of a similar nature. This wide investigation into the computational complexity of the planted clique problem has, however, mostly focused on its time complexity. In this work, we ask- Do the statistical-computational phenomena that make the planted clique an interesting problem also hold when we use `space efficiency' as our notion of computational efficiency? It is relatively easy to show that a positive answer to this question depends on the existence of a O(log n) space algorithm that can recover planted cliques of size k = Omega(sqrt{n}). Our main result comes very close to designing such an algorithm. We show that for k=Omega(sqrt{n}), the recovery problem can be solved in O((log*{n}-log*{k/sqrt{n}}) log n) bits of space. 1. If k = omega(sqrt{n}log^{(l)}n) for any constant integer l > 0, the space usage is O(log n) bits. 2.If k = Theta(sqrt{n}), the space usage is O(log*{n} log n) bits. Our result suggests that there does exist an O(log n) space algorithm to recover cliques of size k = Omega(sqrt{n}), since we come very close to achieving such parameters. This provides evidence that the statistical-computational phenomena that (conjecturally) hold for planted clique time complexity also (conjecturally) hold for space complexity.

中文翻译：

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种植团恢复的空间复杂度和检测的空间复杂度一样吗？

我们研究了种植团问题，其中在 Erd\H{o}sR\'enyi 图 G(n, 1/2) 中种植了一个大小为 k 的团，并且有人对检测或恢复这个种植团感兴趣。这个问题很有趣，因为人们普遍认为它在群大小 k=sqrt{n} 处显示统计计算差距，并且已经成为具有这种差距的原型问题，从中可以推断出其他统计问题的平均难度. 与其他类似性质的问题不同，它还显示了检测和恢复变体之间的紧密计算联系。然而，对种植集团问题的计算复杂性的广泛研究主要集中在其时间复杂性上。在这项工作中，我们问——当我们使用“空间效率”作为我们的计算效率概念时，使种植集团成为一个有趣问题的统计计算现象是否也成立？相对容易证明这个问题的肯定答案取决于 O(log n) 空间算法的存在，该算法可以恢复大小为 k = Omega(sqrt{n}) 的种植团。我们的主要结果非常接近于设计这样一个算法。我们表明，对于 k=Omega(sqrt{n})，恢复问题可以在 O((log*{n}-log*{k/sqrt{n}}) log n) 位空间中解决。1. 如果 k = omega(sqrt{n}log^{(l)}n) 对于任何常数整数 l > 0，空间使用量为 O(log n) 位。2.如果k = Theta(sqrt{n})，空间使用量为O(log*{n} log n)位。我们的结果表明确实存在一个 O(log n) 空间算法来恢复大小为 k = Omega(sqrt{n}) 的团，因为我们非常接近实现这些参数。这提供了证据，证明（推测）适用于种植集团时间复杂度的统计计算现象也（推测）适用于空间复杂度。