802 - Lead or Gold

All about problems in Volume 8. If there is a thread about your problem, please use it. If not, create one with its number in the subject.

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mf
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Re: 802 - Lead or Gold

Post by mf »

Oh, now I see. I knew that trick, I just forgot its name.

But I think its use is totally unnecessary here. In fact, you don't even have to add any artificial variables. Here's LP formulation that I've used in my simplex solution:

Code: Select all

minimize (A+B+C) - \sum_{i=1}^n (a[i] + b[i] + c[i]) x[i]
subject to
\sum_{i=1}^n a[i] x[i] <= A
\sum_{i=1}^n b[i] x[i] <= B
\sum_{i=1}^n c[i] x[i] <= C
x[i] >= 0
That is, I just turn strict equalities into inequalities and try to minimize the sum of residuals, starting from zero basic feasible solution. The original problem is feasible if that sum reaches zero.

serur
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Re: 802 - Lead or Gold

Post by serur »

Also I noticed that in CLR -- without Stein -- they give no description of simplex algo.
I guess you have an English version of CLRS that you brought from one of your innumerable
trips around the world ;)
If there is ever a war between men and machines, it is easy to guess who will start it (c) Arthur Clarke

mf
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Re: 802 - Lead or Gold

Post by mf »

Yes, they've added a chapter on linear programming only in 2nd edition of the book.

By the way, the book's third edition is coming this fall!

serur
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Re: 802 - Lead or Gold

Post by serur »

I got AC with your LP formulation, as well. Of course, your approach is neat:
your just sum the residuals and change the order of summation and get the objective function (just dropping
a constant summand).
If there is ever a war between men and machines, it is easy to guess who will start it (c) Arthur Clarke

tryit1
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Re: 802 - Lead or Gold

Post by tryit1 »

i got AC on 10089 but this problem is giving me trouble when i've vectors like (0,0,q) or (0,q,r).

If all were positive i could convert it to 10089 by multiplying the lcm of (p,q,r) to make the ratio equations equal. But this zero is messing the whole thing. Any tricks , can we add newEPS=1e-3 in place of 0 and actual EPS=1e-9. but adding newEPS to each quantity of that will change the ratio and possibly solution.

mf
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Re: 802 - Lead or Gold

Post by mf »

i got AC on 10089 but this problem is giving me trouble when i've vectors like (0,0,q) or (0,q,r).
You're talking about weights in the desired (target) mixture? Well, if some component is zero, it's just the same problem with one less dimension. If target mixture is (a, b, 0), then the only mixtures you may take are also of form (x, y, 0). So it doesn't hurt to remove all mixtures which are not of this form, drop this 0 coordinate from all vectors, and you get a 2D problem.

(Note: I haven't actually tried to solve this problem by reducing to 10089, so maybe I'm saying here something wrong. I've got accepted with simplex method, and a method based on projection and convex hull.)

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