Friday, February 13, 2009

Combinatorial Complexity

There is a certain broad area of complexity theory which does not have a proper name.

As far as it occurs to me at this instant, there are (at least) three big areas of complexity theory:

1. "Structural complexity theory". This refers to understanding the relationships between complexity classes, bounding and classifying time vs. space vs. randomness, understanding reductions, etc.

2. "Algorithmic complexity theory". I just made this name up, but I'm thinking here about the area of understanding the complexity of various specific problems; I think of proving NP-hardness results as falling into this area, so I would put inapproximability and PCPs into this category.

3. The mystery area I referred to at the beginning of this post. I guess my best attempt at naming this category would be either "Combinatorial complexity", or "Lower bounds". Here I mean the study of things like formula size/depth, constant-depth circuit size, branching program sizes, decision tree size, DNF size... roughly, the study of non-uniform classes of computation that are "small" enough that one can make reasonable progress on proving lower bounds.


The impetus for this post is to point you to a nice-looking graduate course on this 3rd area being taught by Rocco Servedio at Columbia this term. If the lecture notes continue, this should be a great resource for this huge area of complexity theory.

3 comments:

  1. The lecture notes should keep being posted as the scribes finish them.

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  2. Isn't this area sometimes referred to as "concrete complexity" (perhaps motivated by fact that the lower bounds are for concrete, crisp problems)?

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  3. Yes, good point Venkat. I forgot this is sometimes called Concrete Complexity, which is a good name.

    Also, there are way more branches of Complexity Theory :)

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