1

Let's say I have a set of non-negative integers $a_1,..,a_n$ and a number $C$ which is a non-negative constant (an integer).

Consider an equation

$x_1\cdot a_1 + x_2\cdot a_2 + ... + x_n\cdot a_n - C = 0$

where $x_1,...,x_n$ are unknown non-negative integers. I have to say whether a solution to this equation exists, and when does it exist and when does it not exist.

I'd really appreciate anyone's help I've been struggling with this one for quite a while.

EDIT
Well so far i haven't made a good progress, i'm going to sleep and see you all in the morning.So far i got two conclusion. 1.) if the number C is not divisible by the gcd of the set than there certainly is no solution 2.)If C is divisble by gcd, If we divide all the numbers by the gcd than it becomes Frobenius coin problem.I'm now looking into upper bounds for FCP.See you all in the morning.

mikey
  • 67
  • 2
  • I forgot to mention that all the numbers in the Set, along with C are non negative integers. – mikey Dec 05 '12 at 21:28
  • And the $x_i$ are unknown? (Just confirming...) – apnorton Dec 05 '12 at 21:34
  • Yes x's are unknowns. – mikey Dec 05 '12 at 21:35
  • mikey: While formatting your question, I added the "non-negative" qualifications for the $a_i$ and $C$, okay? – amWhy Dec 05 '12 at 21:37
  • Yeah, thanks for that. – mikey Dec 05 '12 at 21:38
  • How much help do you want us to give you here? I could tell you the answer, but you wouldn't learn very much... – Jonathan Christensen Dec 05 '12 at 21:40
  • Well, some hint would be nice, like what was your train of thought while solving it... what did you look at first etc etc...

    This problem is due in tuesday so i've got time no probs :)

     – mikey Dec 05 '12 at 21:42
  • 2
    I would suggest first trying it out for two numbers. – EuYu Dec 05 '12 at 21:47
  • 1
    I give one hint. The umbrella concept is numerical semigroup. It seems to me that in general the full answer is not known (or at least difficult to express), but I may be wrong. Because all the variables must be non-negative it is not just about gcd. – Jyrki Lahtonen Dec 05 '12 at 21:50
  • Ok, here's a hint: what happens if all the $a_i$ are even? What if they're all multiples of $3$, or $6$, or $k$ for any integer $k$ you like? Under what conditions can you find a solution? Then start considering small cases (two variables, like EuYu suggests, or maybe three) that aren't so simple, and see when you can find solutions – Jonathan Christensen Dec 05 '12 at 21:51
  • For an example see this question, where the case $n=2$, $a_1=3$, $a_2=5$ is analyzed in detail. – Jyrki Lahtonen Dec 05 '12 at 21:57
  • looks very like a projection of position vector onto a normal vector to N dimentional surface, beying C it's "distance" to the origin – Mykolas Dec 05 '12 at 21:59
  • A dynamic programming solution (based on knapsack) would work – Jacob Dec 07 '12 at 05:02
  • Will dynamic programming give definitive answers of the type If the numbers $a_i$ have no common divisors, then all sufficiently large $C$ can be written in this form? – Jyrki Lahtonen Dec 07 '12 at 14:10

2 Answers2

1

This way is really inefficient for a large number of variables, but Integer Programming is a built-in feature in most scientific computation software these days.

Try to find whether this Integer Program has a solution

$$\mathbf{min} \sum_ix_i$$ $$s.t. \sum_i a_i x_i = C$$ $$x_i \geq0$$ where all $x_i$ are integers. This will give you a dirty but easy answer.

Note: The objective function $\sum x_i$ is just a placeholder. It can be replaced by any other increasing function. If the software supports it, even $\mathbf{min}\mbox{ } 0$ is fine. Thanks to Jacob for pointing it out.

dexter04
  • 1,971
  • You don't need to minimize anything. The OP is merely asking for a feasible solution to your problem. Also $x_i \ge 0$. – Jacob Dec 05 '12 at 22:26
  • @Jacob: An IP must have an objective function to minimize. Before solving, all solvers check whether it is feasible or not. So, I have put a dummy objective fn – dexter04 Dec 05 '12 at 22:29
  • Not really. $\text{minimize } 0$ is fine. – Jacob Dec 05 '12 at 22:29
  • yeah. that is fine. I was just trying to formulate it as an IP. – dexter04 Dec 05 '12 at 22:33
0

Solve it as a knapsack problem. $a_i$ is your weight, $x_i$ is the quantity of each item and set the "value" for each $x_i$ as 1. $C$ is the total weight your knapsack can carry. If you do find a solution, check if the total "weight" is equal to $C$ ; then you have a solution.

Jacob
  • 2,540