CS432, Fall 2005
Solutions for HW6

Q1:
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This won't be on the exam, so no need to give details (I don't
think).  A Google search will take you the Clay Mathematics 
Institute pages or the Minesweeper work's home-page.

We hope this quick look will convince you that many of these 
odd-seeming NP-c problems crop up in "everyday" situations 
(e.g. maximizing the number of filled rows).  While often the 
probem sizes are small enough that things work out (we *can* 
play Tetris, after all), perhaps it's not a surprise that part 
of what makes many puzzles and certain games (their modern 
computer-based counterparts) hard is something to do with the 
fact that they're based on hard problems (i.e. NP-c problems).

Q2:
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A) What is the difference between a software patent and a copyright? 
Answer:  See content item #1 here:
http://en.wikipedia.org/wiki/Software_patent

B) If Strassen were to have submitted his matrix multiplication
algorithm to the US Patent Office in the last few decades, do you 
think his patent would have been accepted? Say why or why not. 
Answer:  Almost certainly not, since this is a good example of
a "mathematical expression" or algorithm.

Two questions on "captcha":

C-1) As a computer scientist, do you think what has been patented
here is an algorithm that is described clearly and precisely? 
Is it something larger than an algorithm? 
Answer:  Clearly this is larger than one algorithm; it's probably 
a reasonable example of the kind of "process" described in the 
readings (like one-click shopping).  In my opinion, it's 
described reasonably clearly but not described as precisely 
as an algorithm; it's more like a description of a family 
of algorithms.

C-2) Also, in your or your group's opinion, do you think 
governments should patent things like this? Give your reasons. 
(They can be technical ones related to this patent, or broader 
reasons based on philosophy or societal issues.) 
Answer:  Better answers will have some basis obtained from 
what they read (and not just their opinion/feelings).


Q3: Showing 0/1 Knapsack is NP-hard
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As noted in the book, slides and lecture, we need to show a 
polynomial reduction from a known  NP-c (or NP-hard) problem to 
the 0/1 knapsack.  The problem tells you to use SubsetSum 
problem.   So show any input x of SubsetSum can be transformed 
into a valid input T(x) for 0/1-Knapsack, where x will be 
accepted by SubsetSum if and only if T(x) is accepted by 0/1-
Knapsack.

Each of the n items for subset sum have a value, i.e. x_i.  
There's also a target value, k, for subset sum.

The hints try to make you see that if you just need to do this:
1) simply map each item in the subset sum problem to one item 
for 0/1 knapsack.
2) for each item, give both the knapsack value and weight for 
that item the x_i value from the subset sum input.
3) give both W and k for the knapsack the value k from the 
subset sum input.

This is trivially polynomial.  Steps 1/2 above are Theta(n) and 
Step 3 is constant.

Why is this a valid transformation?  Try it on the two examples 
in the problem and you'll see.  The constraint on the weights 
will make sure that the sum of the items is <= the original k 
value.   The constraint on the values will make sure the sum of 
the items is >= the orginal k.  A value can only be *both* >= 
*and* <= some target if is is exactly equal, which is what we 
want.

So in one sense, the subset sum problem is just a simplified 
version of the 0/1 knapsack problem.

And, with this transformation and the dynamic programming 
solution to 0/1 knapsack, you can solve the subset sum problem 
directly!


NOTE:  In HW5, you showed that the 0/1 knapsack problem is in NP.
Here in HW6 you just showed that it's NP-hard, i.e. that any
problem in NP can be reduced to 0/1 knapsack.
So both of these things together prove that 0/1 knapsack is NP-c.