. Find the academic paper by Erik Demaine (and others) that talks about Tetris a

Question

. Find the academic paper by Erik Demaine (and others) that talks about Tetris and NP-Completeness

and answer the following questions:

(a) What problem are they proving is NP-Complete? (Tetris" isn't a complete answer to this

question. I am not asking you to copy the detailed description of how the authors made a decision

problem from the game Tetris. Instead, I want a simple explanation of what yes/no question they

are asking in their Tetris decision problem.)

(b) What NP-Complete problem did they reduce to their problem? Give a small example instance of that problem.

(c) How old was Erik Demaine when that paper was presented at COCOON (an academic conference)?

Explanation / Answer

a)

Check the definitions of NP-completeness and NP-hardness. I suspect they use Karp reductions for one and Cook reductions for the other.

Then, the quote makes sense: it is indeed unknown if Cook reductions and Karp reductions are equivalent in this sense. This is related to the question whether co-NP = NP; Cook reductions can not separate the two but Karp reductions might. See here for some details.

However, the authors use different definitions than (most of) the rest of the CS world: the widely accepted standard is to use Karp reductions.

To be fair, the authors want to talk about optimization problems, in which case they need a notion of (NP-)hardness that is not tied to decision problems. Karp reductions won't get them there directly.

They picked a poor way out, arguably, by defining the notions differently from everybody else. There are cleaner (but more involved) ways, defining complexity classes for optimization problems rigorously, providing them with their own types of reduction, and tying them to the classic decision-problem classes using threshold languages.

b)The subject of computational complexity theory is dedicated to classifying problems by how hard they are. There are many different classifications; some of the most common and useful are the following. (One technical point: these are all really defined in terms of yes-or-no problems -- does a certain structure exist rather than how do I find the structure.)

NP does not stand for "non-polynomial". There are many complexity classes that are much harder than NP.

Although defined theoretically, many of these classes have practical implications. For instance P is a very good approximation to the class of problems which can be solved quickly in practice -- usually if this is true, we can prove a polynomial worst case time bound, and conversely the polynomial time bounds we can prove are usually small enough that the corresponding algorithms really are practical. NP-completeness theory is concerned with the distinction between the first two classes, P and NP.

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