There are two aspects to this question - the core of the question is, can problem constraints change the asymptotic complexity of an algorithm? The answer to that is yes. But then you give an example of a constraint (strings limited to 15 characters) where the answer is: the question doesn't make sense. A lot of the other answers here are misleading because they address only the second aspect but try to reach a conclusion about the first one.

Formally, the asymptotic complexity of an algorithm is measured by considering a set of inputs where the input sizes (i.e. what we call n) are unbounded. The reason n must be unbounded is because the definition of asymptotic complexity is a statement like "there is some n₀ such that for all n ≥ n₀, ...", so if the set doesn't contain any inputs of size n ≥ n₀ then this statement is vacuous.

Since algorithms can have different running times depending on which inputs of each size we consider, we often distinguish between "average", "worst case" and "best case" time complexity. Take for example insertion sort:

• In the average case, insertion sort has to compare the current element with half of the elements in the sorted portion of the array, so the algorithm does about n²/4 comparisons.
• In the worst case, when the array is in descending order, insertion sort has to compare the current element with every element in the sorted portion (because it's less than all of them), so the algorithm does about n²/2 comparisons.
• In the best case, when the array is in ascending order, insertion sort only has to compare the current element with the largest element in the sorted portion, so the algorithm does about n comparisons.

However, now suppose we add the constraint that the input array is always in ascending order except for its smallest element:

• Now the average case does about 3n/2 comparisons,
• The worst case does about 2n comparisons,
• And the best case does about n comparisons.

Note that it's the same algorithm, insertion sort, but because we're considering a different set of inputs where the algorithm has different performance characteristics, we end up with a different time complexity for the average case because we're taking an average over a different set, and similarly we get a different time complexity for the worst case because we're choosing the worst inputs from a different set. Hence, yes, adding a problem constraint can change the time complexity even if the algorithm itself is not changed.

However, now let's consider your example of an algorithm which iterates over each character in a string, with the added constraint that the string's length is at most 15 characters. Here, it does not make sense to talk about the asymptotic complexity, because the input sizes n in your set are not unbounded. This particular set of inputs is not valid for doing such an analysis with.

In the mathematical sense, yes. Big-O notation describes the behavior of an algorithm in the limit, and if you have a fixed upper bound on the input size, that implies it has a maximum constant complexity.

That said, context is important. All computers have a realistic limit to the amount of input they can accept (a technical upper bound). Just because nothing in the world can store a yottabyte of data doesn't mean saying every algorithm is O(1) is useful! It's about applying the mathematics in a way that makes sense for the situation.

Here are two contexts for your example, one where it makes sense to call it O(1), and one where it does not.

1. "I decided I won't put strings of length more than 15 into my program, therefore it is O(1)". This is not a super useful interpretation of the runtime. The actual time is still strongly tied to the size of the string; a string of size 1 will run much faster than one of size 15 even if there is technically a constant bound. In other words, within the constraints of your problem there is still a strong correlation to n.
2. "My algorithm will process a list of n strings, each with maximum size 15". Here we have a different story; the runtime is dominated by having to run through the list! There's a point where n is so large that the time to process a single string doesn't change the correlation. Now it makes sense to consider the time to process a single string O(1), and therefore the time to process the whole list O(n)

That said, Big-O notation doesn't have to only use one variable! There are problems where upper bounds are intrinsic to the algorithm, but you wouldn't put a bound on the input arbitrarily. Instead, you can describe each dimension of your input as a different variable:

n = list length
s = maximum string length
=> O(n*s)

It depends.

If your algorithm's requirements would grow if larger inputs were provided, then the algorithmic complexity can (and should) be evaluated independently of the inputs. So iterating over all the elements of a list, array, string, etc., is O(n) in relation to the length of the input.

If your algorithm is tied to the limited input size, then that fact becomes part of your algorithmic complexity. For example, maybe your algorithm only iterates over the first 15 characters of the input string, regardless of how long it is. Or maybe your business case simply indicates that a larger input would be an indication of a bug in the calling code, so you opt to immediately exit with an error whenever the input size is larger than a fixed number. In those cases, the algorithm will have constant requirements as the input length tends toward very large numbers.

From Wikipedia

Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity.
...
In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows.

In practice, almost all inputs have limits: you cannot input a number larger than what's representable by the numeric type, or a string that's larger than the available memory space. So it would be silly to say that any limits change an algorithm's asymptotic complexity. You could, in theory, use 15 as your asymptote (or "particular value"), and therefore use Big-O notation to define how an algorithm grows as the input approaches that size. There are some algorithms with such terrible complexity (or some execution environments with limited-enough resources) that this would be meaningful.

But if your argument (string length) does not tend toward a large enough value for some aspect of your algorithm's complexity to define the growth of its resource requirements, it's arguably not appropriate to use asymptotic notation at all.

NO!

The time complexity of an algorithm is independent of program constraints. Here is (a simple) way of thinking about it:

Say your algorithm iterates over the string and appends all consonants to a list.

Now, for iteration time complexity is O(n). This means that the time taken will increase roughly in proportion to the increase in the length of the string. (Time itself though would vary depending on the time taken by the if statement and Branch Prediction)

The fact that you know that the string is between 1 and 15 characters long will not change how the program runs, it merely tells you what to expect.

For example, knowing that your values are going to be less than 65000 you could store them in a 16-bit integer and not worry about Integer overflow.

Do problem constraints change the time complexity of algorithms?

No.

If I know for sure that the length of the string is less than, say, 15 characters ..."

We already know the length of the string is less than SIZE_MAX. Knowing an upper fixed bound for string length does not make the the time complexity O(1).

Time complexity remains O(n).

A fundamental requirement of big-O notation is that parameters do not have an upper limit. Suppose performing an operation on N elements takes a time precisely equal to 3E24*N*N*N / (1E24+N*N*N) microseconds. For small values of N, the execution time would be proportional to N^3, but as N gets larger the N^3 term in the denominator would start to play an increasing role in the computation.

If N is 1, the time would be 3 microseconds.
If N is 1E3, the time would be about 3E33/1E24, i.e. 3.0E9.
If N is 1E6, the time would be about 3E42/1E24, i.e. 3.0E18
If N is 1E7, the time would be 3E45/1.001E24, i.e. ~2.997E21
If N is 1E8, the time would be about 3E48/2E24, i.e. 1.5E24
If N is 1E9, the time would be 3E51/1.001E27, i.e. ~2.997E24
If N is 1E10, the time would be about 3E54/1.000001E30, i.e. 2.999997E24

As N gets bigger, the time would continue to grow, but no matter how big N gets the time would always be less than 3.000E24 seconds. Thus, the time required for this algorithm would be O(1) because one could specify a constant k such that the time necessary to perform the computation with size N would be less than k.

For any practical value of N, the time required would be proportional to N^3, but from an O(N) standpoint the worst-case time requirement is constant. The fact that the time changes rapidly in response to small values of N is irrelevant to the "big picture" behaviour, which is what big-O notation measures.

Big-O measures the complexity of algorithms, not of code. It means Big-O does not know the physical limitations of computers. A Big-O measure today will be the same in 1 million years when computers, and programmers alike, have evolved beyond recognition.

So restrictions imposed by today's computers are irrelevant for Big-O. Even though any loop is finite in code, that need not be the case in algorithmic terms. The loop may be finite or infinite. It is up to the programmer/Big-O analyst to decide. Only s/he knows which algorithm the code intends to implement. If the number of loop iterations is finite, the loop has a Big-O complexity of O(1) because there is no asymptotic growth with N. If, on the other hand, the number of loop iterations is infinite, the Big-O complexity is O(N) because there is an asymptotic growth with N.

The above is straight from the definition of Big-O complexity. There are no ifs or buts. The way the OP describes the loop makes it O(1).

It will be O(1) i.e. constant.

This is because for calculating time complexity or worst-case time complexity (to be precise), we think of the input as a huge chunk of data and the length of this data is assumed to be n.

Let us say, we do some maximum work C on each part of this input data, which we will consider as a constant.
In order to get the worst-case time complexity, we need to loop through each part of the input data i.e. we need to loop n times.
So, the time complexity will be:
n x C.

Since you fixed n to be less than 15 characters, n can also be assumed as a constant number.
Hence in this case:
n = constant and,
(maximum constant work done) = C = constant
So time complexity is n x C = constant x constant = constant i.e. O(1)

Edit

The reason why I have said n = constant and C = constant for this case, is because the time difference for doing calculations for smaller n will become so insignificant (compared to n being a very large number) for modern computers that we can assume it to be constant.

Otherwise, every function ever build will take some time, and we can't say things like:
lookup time is constant for hashmaps