bounds in the model of communication complexity. Maybe the analogy is helpful.

I am thinking the multiparty number-on-the-forehead

model of communication complexity. In this model k-players wish to evaluate

a Boolean function where each is a n-bit string. In what follows

think of the output of f as being or . Player

receives as

input all of the except (which is “written on his forehead”). The question is

how many bits the players need to communicate in order to evaluate the function.

It is usually assumed that the communication is broadcast and known to all players.

So the required communication will be a number between 0 and n.

The norm that has been used in this setting is quite similar in spirit to the norm. The “k”

version of the norm is applied to a k-argument function f and is defined as follows.

Babai, Nisan, and Szegedy showed that if a function has small such norm, then it requires a lot

of communication. To show that f is hard, it also suffices to find some g such that g correlates with

f and g has small norm—in other words, one can lower bound the dual norm .

More precisely, the way this is applied to the communication problem is that

where is a normalization factor I have hopefully gotten correct, and is the k-party communication complexity.

The “set intersection” function, determining if there is a position where

all the share a 1, is an example of a function where the

norm is relatively large, yet one can still show a strong lower bound via the dual norm approach.

The main drawback with this technique is that it gives lower bounds of

size at most , and so becomes trivial for many players. This is where

things start to get most interesting for applications to circuit complexity. Combining a result of Hastad-Goldmann with a circuit simulation result of Beigel-Tarui (building on Yao and Allender), any function which can be

computed by a polynomial size, constant depth circuit with AND, NOT and mod m gates

has a number-on-the-forehead protocol with polylog many players and polylog communication.

Needless to say, we currently don’t know of any explicit function hard for this class of circuits (even

just with mod 6 gates).

Slightly related (But in the AC_0 shallower water): I think it is a very interesting problem to show that the Boolean function f describing the property: “a graph on n vertices has a clique of size 2 log n” cannot be approximated by a function in AC_0 (a bounded depth polynomial size Boolean circuit). It is known that f itself is not in AC_0 (Paul Beame was the first to show it I think). Of course depth 2 suffices if you allow quasi polynomial circuits. (Actually I am not aware of any function computable in quasi-polynomial size bounded depth circuits where it is known that the function cannot be approximated by a function in AC_0.)

]]>Sorry, I didn’t really answer your question properly. What I mean is that ðŸ™‚ would be ecstatic to find a quasipolynomial function that didn’t have a polynomial-time algorithm, even if finding a superquasipolynomial lower bound for an NP function would be more intellectually satisfying.

I presume the algorithm you refer to is based on showing that if you know there’s a clique of size and you find all the cliques of size then almost certainly their union (or something pretty close to their union) is the clique of size Is that roughly the idea? If you changed the probability so that the expected size of the biggest clique was say, would that make it much harder to find a clique of size ?

]]>That’s a good question. Ideally, ðŸ™‚ would want to prove that PNP, but by this stage of the conversation *anything* new and exciting would do. However, there seems to be little prospect of that (even by the end of the tenth instalment).

I still think there might be a more realistic Polymath project in there somewhere. Perhaps when I get to the end I’ll collect together all the questions that seem interesting and replace the old proposal 5 by a set of proposals 5a to 5d (or whatever letter I get up to).

And thanks for drawing my attention to that paper — I look forward to checking it out.

]]>— also has a nice survey of the state of the art, including his own work showing easiness results (for ) and hardness results (for ) even in the semirandom model, where an adversary can delete any edges it wants from a random graph + planted (except for edges from the planted clique, of course).

On a more philosophical note… You can of course find a clique of size in time, which is only quasipolynomial in if is logarithmic (or polylogarithmic). Indeed, as Coja-Oghlan points out, Alon-Krivelevich-Sudakov can find a planted -clique in a random graph in time for any .

For the purposes of lower bounds, one often (but not always) does not make a big distinction between polynomial and quasipolynomial time/size. (We don't believe SAT is solvable even in time, let alone time.)

Are the characters in the dialogue suggesting proving a lower bound for problems which are solvable by quasipolynomial size circuits? Or would the optimist ðŸ™‚ hope to separate NP from quasipolynomial-size circuits as well?

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