Integrality gap in Online Problems and adaptation to competitive ratio

As we all know, in offline problems it is common practice to calculate the integrality gap to get some bound on the approximation ratio of the integral solution.

Now this gap ($ IG:=\frac{OPT_{frac}}{OPT_{int}}$ ) relies on the fact that we can compute the fractional optimum by just solving the relaxed linear program of the problem.

In Online Algorithms there is no LP which can be relaxed and solved. Well there technically is but we can’t solve it like we do in the offline setting, since online algorithms must make decisions irrevocably and without knowing what to come.

So maybe we are lucky and we can approximate a fractional solution online (by a factor of $ \alpha$ ) and round this solution with the loss of the $ IG$ (also online) to get some integral solution for our problem.

My question is, how we can adapt the knowledge of our offline setting to the online setting.

As a matter of fact, can we just conclude that if we have a online $ \alpha$ -approximation for the fractional online problem, that we could (at most) have some $ \frac{\alpha}{IG}$ -approximation by rounding.

Thanks in advance

Number partitioning targeting ratio of subset sums and equal size

I’ve seen a number of questions and answers related to the partitioning problem of dividing a set into 2 subsets of equal size and sum that use greedy or dynamic programming solutions to get approximate answers. However, I am looking to split a set into 2 subsets of minimum difference in size but with a target ratio of sums.

I have tried variations of greedy algorithms that will work to minimize the largest difference of the two metrics at any point in the calculation, but this just results in a result that splits the difference between the two. I’m happy with an approximate solution since it needs to run in a reasonable amount of time.

Calculate hit ratio

Consider the following code:

int[] A;

A = new int [1024]; // allocating memory for 1024 integers.

int sum = 0;

for ( i = 0; i < 1024; i++)

      sum = sum + A[ i ]; 

Give the following:

There are two level of memory: cache and main memory.

The size of the program is 2 KB and resides in cache memory before the program starts execution.

The int type is 4 bytes long.

The operating system allocates only 128 bytes of cache for the data (i.e. for array A)

What is the hit ratio for this program?

How to display an infinite ratio to best communicate that the set contains exclusively one type?

I have a reporting dashboard displaying a ratio of apples to oranges normalized to 1:x such as 10 apples and 20 oranges is “1:2”.

How would I best display 0 apples 20 oranges?

  • “1:Infinity” is an option but looks weird
  • “0:20” or “0:1” shouldn’t be options due to the requirement of normalizing it in the form of “1:x”, but really the intent is to communicate to the user that for this cell of the table there are “only oranges”

Keyword Golden Ratio KGR Keyword Super Profitable Keyword Method

Any one have any experience about Keyword Golden Ratio or KGR Keyword method ? It's developed by Doug Cunnington. It's a method to finding super profitable and easy ranking keywords. These keywords could rank even without any backlink with in a month. KGR keywords could bring more sales than general. But it hard to find…

Keyword Golden Ratio KGR Keyword Super Profitable Keyword Method

Would It be Useful To Add a Column Of Success:Failed:Wrong Ratio In CAPTCHA Stats?

Hello, all! Hope everyone has had a productive almost over!~
I am checking out CATPCHA stuff on SER again. Left too much for too long alone w/o really checking or making improvments based on feedback of what I’m doing = Poor Experimentation.
Anyway, Sven, my question is whether such a feature as this (as in title) would be useful?
I think it might be, as a SER user could then easily see how productive each service is without having to either calculate in their heads or use another method, thus saving time and giving a clearer comparative picture between them all.