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I am trying to show the existence of a polynomial size, bounded depth monotone circuit on the inputs $(x_1,\ldots, x_n)$ that gives $1$ if $\sum x_i \geq n/2 + n/\log n$ and $0$ if $\sum x_i \leq n/2 - n/\log n$. This is a modified form of the MAJORITY problem.

I want to use the probabilistic method but I cant seem to get it to work. I found a way to get a circuit that was not of bounded depth, iterate a bunch of random layers of gates that compute the MAJORITY function for three randomly selected inputs. This will, for $\mathcal{O}(\log n)$ depth, give the right result with overwhelming probability, so one such circuit is always correct.

I think I should have layers of alternating AND and OR gates, but since the $n/\log n$ factor is so small, even when we should get a $1$ for instance, the probability that a random input is a $1$ is only $1/2 + 1/\log n$, not really so high. I cannot seem to amplify this enough to take care of both cases.

Any help?

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This task is known as approximate counting or approximate majority. The first to construct such a circuit was Ajtai in his celebrated 1983 paper $\Sigma_1^1$-formulae on finite structures, which also proves the first switching lemma. You can check out Section 3 for the construction. For a more modern account, see Section 2 of Viola's Randomness buys depth for approximate counting.

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  • $\begingroup$ Thanks, these papers are going a bit above my head I am afraid. However, from reading some of Viola's paper, isn't he actually saying in the abstract that this problem cannot be solved in depth $2$ polysize circuits, but this does not concern me as the depth should be bounded but may be large? $\endgroup$ – Slugger Jul 21 at 23:33
  • $\begingroup$ Section 2.1 of Viola's paper gives a construction for $n/2 \pm n/\log^{d-1}n$, which uses depth $d$. $\endgroup$ – Yuval Filmus Jul 21 at 23:36
  • $\begingroup$ You could try using the keywords "approximate counting" or "approximate majority" to find other expositions of this construction. I'm afraid all constructions would be somewhat technical. $\endgroup$ – Yuval Filmus Jul 21 at 23:37
  • $\begingroup$ Thanks for the replies! My knowledge is more in the areas of combinatorics and graph theory and not so much theoretical computer science so I am not really familiar with much of the terminology in these papers. Anyway, I will have a look :) $\endgroup$ – Slugger Jul 21 at 23:40

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