эта известная проблема="this well-known problem", это="it is" (это - человек) or neuter "this" (это имя) or indefinite "it" (я вижу это)
surely saying "it's a common problem" is more realistic than saying "it's a well-known/famous
A problem can be very common, but almost unknown e.g. a fault in a model of washing machine that has not yet been recalled.
The meaning is "important problem"? I'm not a native speaker of English but "well-known problem" sounds quite weird to me.
To say "well-known problem" does sound kind of strange, but it could be possible under certain circumstances. "Well-known issue" is a better collocation in English. I think the translation here is simply showing that the Russian word известный can be translated as well-known.
This is perfectly natural to me (Hiberno- and British-English speaker), but that may be because I am a mathematician! There are many famous problems, or well-known problems, in mathematics and theoretical physics, some of which have $1,000,000 prizes!
Being a computer scientist, I agree:
"The list below contains some well-known problems that are NP-complete when expressed as decision problems."
I take notice though of the native speakers comment that "Well-known issue" might be more natural when the problems are not mathematical in nature but a "real world issue".
Do you think "well-known problem" would be a phrasing used by better speakers? Simple folk might say "everyone knows it" :p
Does the word "извесная" have anything to see with the word "извините" ? 'cause they look very much alike ^^
Известная and извините look alike only because they have the same prefix из- and the same first root letter в. Taking the root we have "вест" and "вин" which are clearly different. Весть=a piece of news, вина=guilt.