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Topic: On NP Completeness and Intractability
Replies: 90   Last Post: Nov 1, 2005 8:08 PM

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 tchow@lsa.umich.edu Posts: 1,133 Registered: 12/6/04
Re: On NP Completeness and Intractability
Posted: Oct 27, 2005 7:26 PM

In article <djrln4\$f8l\$1@joe.rice.edu>, Randy <joe@burgershack.com> wrote:
>Interesting. I'd never thought of each step in a game as being a
>problem in its own right. But is that a decision problem?

It's easy to cast, say, chess as a decision problem. For example: Given
a position, is it a forced win for White? This is a yes-no question.
If I have the ability to answer this question, then I can play Black and
ensure that I never lose a game in which I have a forced win or draw,
as follows. Whenever it's my turn, I hypothetically try each of my
possible moves (there are only polynomially many such) and ask "forced
win for White?" for each of the resulting positions. Then I pick a move
(if any) for which the answer comes back "no."

By the way, informal reasoning may be helpful here: obviously, when
I'm playing a game, I'm faced with a constant sequence of "decisions"
(what would be a winning move for me to play now?) in an informal sense,
so it should not be surprising that I can formally collect all the
possible situations I might be faced with, and define a decision problem
accordingly.

>I *think* I remember that any optimization problem can be converted into
>a decision problem.

This is true in practice (I'm not sure offhand if one can construct
pathological cases where this fails). Let's say that the optimization
problem is to find, for any given instance, a solution of least cost
(where cost is assumed to be nonnegative). Then the associated decision
problem is, "Given an instance X and a threshold T, does there exist a
solution whose cost is at most T?" In most cases, if you can answer
this question, you can compute the optimum cost by a bisection algorithm.
For example, you might be able to compute a poor but feasible solution
easily and find that its cost is (say) 2^20. Then ask, does there exist
a solution with cost <= 2^19? If yes, ask if there's a solution with cost
<= 2^18, and so on.

Typically you can even construct a solution (not just find the best
cost) if you can solve an associated decision problem. For example,
suppose I want to color a graph with the minimum number of colors.
The associated decision problem would be, "Given a partially colored
graph G and a threshold T, can the partial coloring be completed to a
full coloring using at most T colors?" Then you can color the graph one
node at a time, trying each possible color and picking one that your
infallible decision maker says is O.K. (Note that you never need to
backtrack if the decision maker is indeed infallible.)
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

Date Subject Author
10/25/05 nimmi_srivastav@yahoo.com
10/25/05 Dr. David Kirkby
10/25/05 ianparker2@gmail.com
10/25/05 Pubkeybreaker
10/25/05 osmium
10/25/05 Pubkeybreaker
10/25/05 Willem
10/25/05 Bryan Olson
10/25/05 Willem
10/25/05 Ed Prochak
10/25/05 Dr. David Kirkby
10/25/05 Gene
10/25/05 MartDowd
10/25/05 RobertSzefler
10/25/05 Dr. David Kirkby
10/25/05 Pubkeybreaker
10/25/05 Wim Benthem
10/26/05 RobertSzefler
10/25/05 KP Bhat
10/25/05 stephen@nomail.com
10/25/05 tchow@lsa.umich.edu
10/25/05 googmeister@gmail.com
10/25/05 nimmi_srivastav@yahoo.com
10/25/05 stephen@nomail.com
10/26/05 tchow@lsa.umich.edu
10/27/05 stephen@nomail.com
10/25/05 stephen@nomail.com
10/25/05 Randy
10/25/05 C6L1V@shaw.ca
10/25/05 stephen@nomail.com
10/25/05 Wim Benthem
10/25/05 stephen@nomail.com
10/25/05 Randy
10/25/05 stephen@nomail.com
10/26/05 tchow@lsa.umich.edu
10/26/05 Randy
10/27/05 Arthur J. O'Dwyer
10/27/05 Randy
10/27/05 Willem
10/27/05 Randy
10/27/05 stephen@nomail.com
10/27/05 Randy
10/27/05 stephen@nomail.com
10/27/05 Randy
10/27/05 stephen@nomail.com
10/27/05 tchow@lsa.umich.edu
10/27/05 Randy
10/27/05 tchow@lsa.umich.edu
10/27/05 Joe Hendrix
10/27/05 Randy
10/27/05 Joe Hendrix
10/27/05 Randy
10/27/05 stephen@nomail.com
10/27/05 Arthur J. O'Dwyer
10/27/05 googmeister@gmail.com
10/27/05 tchow@lsa.umich.edu
10/27/05 Randy
10/27/05 Arthur J. O'Dwyer
10/28/05 Joe Hendrix
10/28/05 Ben Rudiak-Gould
10/28/05 Bryan Olson
10/28/05 Pekka Orponen
10/27/05 tchow@lsa.umich.edu
10/27/05 Randy
10/25/05 tchow@lsa.umich.edu
10/25/05 Michael J. Hennebry
10/25/05 stephen@nomail.com
10/25/05 Randy
10/25/05 stephen@nomail.com
10/25/05 Willem
10/25/05 Arthur J. O'Dwyer
10/25/05 googmeister@gmail.com
10/31/05 Oliver Wong
10/31/05 Robert Israel
11/1/05 Willem
11/1/05 googmeister@gmail.com
11/1/05 Willem
11/1/05 Torkel Franzen
11/1/05 googmeister@gmail.com
11/1/05 Oliver Wong
11/1/05 Willem
11/1/05 Robert Israel
10/26/05 Randy
10/26/05 stephen@nomail.com
10/26/05 Randy
10/27/05 Bryan Olson
10/27/05 Randy
10/27/05 Bryan Olson
10/27/05 Randy
10/25/05 Pat Farrell
10/25/05 Bryan Olson