On the usage of "the" article with acronyms and initialisms

I've read Using the definite article with acronyms and initialisms and Is it appropriate to use "the" before an abbreviation? I know that

1. The article is often used with initials that are pronounced letter by letter (initialisms).

2. The article is often NOT used with initials that are pronounced as a word (acronyms).

I used 'often' because I'm not sure about possible exceptions.

In my field of study (Operations Research), initials are frequently used to represent optimization problems (e.g., the TSP). I discussed this point with my university supervisor who makes comments on my reports (English is not his native language). He responded that:

"What I learned from [...] was that whenever we use (specific) acronyms, we don't use 'the' when they are within a sentence, and we use 'the' when a sentence starts with them. For instance,

(i) The HCP is an NP-Complete problem.

(ii) We developed a new algorithm to solve HCP."

HCP stands for the Hamiltonian Cycle Problem. It is really confusing me because I do not know when to use and when not to use the article. For example, on this site, the author Jane Watson writes:

Use a definite article with an initialism if the spelled out term begins with “the” but is not covered in the initialism.

which implies that I should use the article with "HCP". What do you think?

Sadly What I learned from [...] was that whenever we use (specific) acronyms, we don't use 'the' when they are within a sentence, and we use 'the' when a sentence starts with them indicates rather more about his use of it, than about English.

Is it not wholly irrelevant whether the acronym starts or is within a sentence? If the full-out phrase, in this case the Hamiltonian Cycle Problem needs an article then so does the abbreviation… except where the author can demonstrate how that article is included in the abbreviation.

(i) The HCP… seems correct by itself; is an NP-Complete problem contributes nothing.

(ii) … to solve HCP seems to be lacking an article, whichever that be. Please consider a rather different example of MP meaning Member of Parliament.

The crucial point here is that Joe Soap, who is MP works only in a very few, very specific circumstances; otherwise it fails solely because it has no article.

Joe Soap, who is MP for Piddlington does work in both abbreviated and full forms.

Joe Soap, who is the MP for Piddlington does work in both abbreviated and full forms.

Joe Soap, who is an MP does work - even though using an for the abbreviation almost contradicts the ordinary use of a for the full version.

Joe Soap, who is a MP doesn’t work in abbreviation even tough it should work with the full form, Member of Parliament.