Homework Assignment 1

- Think of a problem in NP. Describe carefully the input, the possible
solutions, and why it is easy to check whether a solution is correct.
Specific application-type problems preferred to abstract theory-type
ones. If you're feeling uninspired, try the
compendium of NP optimization problems or some of the puzzles
from my page of
complexity
theory for games and puzzles.
It's ok if your problem does not meet the strict definition of being in NP, as long as it has the same form: a yes/no problem such that a yes answer means that there is some kind of solution which can be quickly checked for correctness, even though it might be difficult to find the solution.

- Describe how to translate instances of your problem into instances
of one of the problems described in class: constraint satisfaction,
maximum independent set, or traveling salesman.
(Note that this is the opposite direction of translation from what you would use to prove your problem NP-complete.)

- What is (are) the natural measure(s) of problem size for your problem? What size instances does your translation produce, as a function of the input size? If the constraint satisfaction (or maximum independent set) problem takes time c^n to solve, how much time would it take to solve your original problem using this translation and a CSP (or MIS) algorithm?

- Finding moving patterns in Conway's Game of
Life.
Input: numbers W,L,S,P.

Output: "yes" if there exists a pattern that repeats itself P generations after the start, shifted over by S steps, and that fits within a W*L rectangle.To check a solution, simply compute the evolution of the pattern for P steps, and compare the result with a shifted copy of the original pattern. (This problem is not actually in NP, because the input size is so small that even writing down a solution is exponential in the input size.)

- We translate to an instance of CSP.
Form a binary variable for each pair (c,t), where c is a cell in the W*L rectangle and t is a time step from 0 to P-1. Form a constraint for each such variable, saying that the variable should match the result of evolving the nine neighboring cells in time step (t-1), or (if t=0) that it should match the result of evolving the nine cells surrounding the point S steps away from its position in step P-1. Also add one more constraint, that at least one cell is nonempty.

- The natural measures are simply the numbers W, L, and P. The translated CSP has W*L*P variables, so the time would be c^{W*L*P}.