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Analysis of an approximation claim

Consider the load balancing problem on two machines. Thus we want to distribute a set of $ n$ jobs with processing times $ t_1,…,t_n$ over two machines such that the makespan (maximum of the processing times of the two machines) is minimized. Professor Smart has designed an approximation algorithm Alg for this problem, and he claims that his algorithm is a $ 1.05$ approximation algorithm. We run Alg on a problem instance where the total size of all jobs is $ 200$ , and Alg returns a solution whose makespan is $ 120$ .

(i) Suppose that we know that all job sizes are at most $ 100$ . Can we then conclude that professor Smart’s claim is false?

(ii) Same question when all job sizes are at most $ 10$ .

Let’s talk about the case (i):

We know that $ \sum{t_i} = 200$ and that $ t_i \leq 100$ . The makespan of the Algorithm $ Alg = 120$ , so $ Alg \leq 1.05 * OPT$ . We have no other information about the algorithm used. A lower bound would be $ LB = max( \frac{\sum{t_i}}{2}, max(t_i)) = max (100,100) = 100$ so I would say for that particular instance we’d have $ 120 \leq 1.05 * 100 = 105$ which means the claim would be false.

Likewise for the case (ii).

My answer is marked as incorrect, and I am struggling to do the right analysis.

Can anyone help please ?

Should it be a claim, a role or a policy?

The distinction between roles and claims is that roles describe a set of users and claims describe a property of a user. So there can be a role “Administrator”, but there can also be a claim “HasElevatedPrivilegeBadge”. Both can allow the same action. Now which one should I pick if I want to allow only certain people to do certain things, for example:

CanAddItem, CanUpdateItem, CanDeleteItem, CanAddProduct, CanUpdateProduct, CanDeleteProduct 

I could create role “Administrator” and add to it claims “CanAddItem”, “CanUpdateItem”, etc., but “CanAddItem” doesn’t describe a property of a user. It says what the user can do, which is not what a claim should do, should it?

Another approach is to create policies, such as:

policy.AddPolicy("CanAddItem", policy => {     policy.RequireAuthenticatedUser()           .RequireRole("Administrator"); }); 

But for more than 20 policies, it will take a good chunk of my Startup class. Is there any other way of doing this, or is one of these the preferred one?

I’d like to point out that I’m specifically looking for a solution for .Net Core Identity. I’m asking for a solution on how to fit my requirement into Identity tables provided by the framework.

For a UK rail delay, which train company do you claim compensation from?

For those who don’t know, the UK rail network was largely privatised in the 1990s. Whether or not rail privatisation is a good thing is still very much up for debate, but even most pro-privatisation people feel the way it was done wasn’t the right way! One upshot is that most non-trivial journeys will involve travel on trains operated by more than one train company.

Let’s consider a hypothetical journey, A to D with a change of trains at C. The A-C journey is to be operated by TrainCoA, C-D by TrainCoB. Unfortunately, the A-C train is cancelled, so you end up getting a later train A-B, another B-C, and then a later train than you’d planned C-D. Oh, and then the C-D train you eventually caught was very late too. Everything was bought on a single ticket.

In this situation, which train company do claim compensation from? TrainCoA, as their initial cancellation was your first delay? TrainCoB, as their C-D delay was the largest delay on your journey? TrainCoC, who ran the A-B journey that you got instead, as they were the first company who’s trains you managed to take? Or can you take advantage of the privatised nature of the system, review the compensation policies of all three train operating companies, and apply for compensation from whichever company has the most generous compensation for your sequence of delays? (Many of the companies have different minimum delays before compensation is due, and different exclusions on what their delay compensation covers….)

Taking photos or pictures of declare and nothing to declare on airport from a relatively far distance in a baggage claim

Hello there 3 days ago at bergen airport i warned my cousin who unfortunately had weird hobbies not to take pictures at the baggage claim since the baggage claim in bergen airport is really close to the custom signs however from a distance he took a picture of the sign declare or nothing to declare from the seats in the baggage claim but he turned off his phone when passing through the customs and put in his pockets. I wonder if it is allowed taking pictures in the baggage claim and signs of the customs area from the baggage claim? because it is not allowed to put our phone out and taking pictures whilst passing the customs or immigration area of an airport.

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Why did my download claim to only take 4.6 GB, and the file itself takes up 4.9 GB?

I am selling my old laptop and I want to put Windows 10 on it since it should sell better that way.

I downloaded Windows 10 using the chromium web-browser and it said the file was only 4.6 GB. I have a DVD-RW that has 4.7 GB of space, but when I check the properties tab of the file, it says the file takes up 4.9 GB. My disk-writing software, brasero, refuses to burn the file to the disk because the file is too large.

Why did the file say it was only 4.6 GB when I downloaded it? Is there something I can do to install Windows 10 on my Ubuntu 18.04 laptop?

I cannot use a flash drive because the USB ports on the laptop are bent.