Several people have mentioned that sometimes a person is just looking to vent, and if so they should say so up front. I agree with that, but even when the problem-haver is explicitly looking for help finding a solution and has stated that up front, even then the partner should listen for the full 120 seconds until the problem has been laid out completely. It’s the interrupting that’s the problem, not the attempt to be a problem-solver.
Some people want to be helped. Some people want to be heard. The trick is finding out which group someone falls in. 
We have a winner! This is what my husband and I have learned to do and it’s wonderful. Expectations are clearly stated and easily met. Of course, this doesn’t work if one person expects the other person to be a mind reader and “just know” what they want.
I’ve actually gotten to the point where I’ll occassionally ask my wife a couple of sentences in “is this a story where you’re looking for some input from me or something you just feel like sharing?” If it’s input, I’ll interrupt to ask questions or to suggest a solution when one comes to me. If it’s that she just needs to vent, then she just needs to vent. I know that nothing I suggest will substitute for her just letting it out.
Which is my wife’s attitude and one I’ll never, ever understand.
If one is not looking for a solution, then why bother to talk? I’ve been told that it “makes one feel better”, but how much better can one feel with a still-unresolved problem?
IMHO: If you’re just whining, be quiet. If you want to actually solve this, then come to me.
But yeah, to go along with what you say, I just let her complain and take 4 times the amount of time needed to perform whatever task she was complaining about.
Because time is short and had you bothered to listen the first time I tried to explain X, you wouldn’t be complaining about it again.
Why does anybody feel the need to complain with no resolution? What good does that do?
It seems your gender is implied in your username, WordMan. 
Yeah, I get that I am just saying that this situation isn’t just an automatic “men are from Mars; women are from Venus” thing - it is more about communication and respect in general, regardless of gender.
I prefer the up-front “is this listening or solving?” type of query. Unfortunately, sometimes things play out too fast to stop and pull back and ask - and sometimes the Problem-Haver isn’t sure which it is…
Again - thanks for all of the inputs thus far…
Sounds like you have a really specific situation in mind that may or may not have anything to do with the situation that prompted the OP.
Sometimes, time is NOT short (and besides, the OP says the venter is only taking a minute or two), and describing the problem to another person is one of the most efficient ways of working through it and deciding on your course of action.
Naw, that’s just my general attitude about everything. Time is short! Get 'er done! No time to waste!
Anyway, my focus isn’t really “hurry up and finish before I interrupt” 'cause interrupting is rude, it’s the more general issue of “why talk about a problem if you’re not looking for solutions?”
Quite a bit, I think. Some people just want to be heard and validated. I think that all people do, at various times.
Not even always validated, either; sometimes the problem that actually needs to be solved isn’t, “what would fix this?” but “am I right to be upset about this?” And for that, offering a solution to the surface problem without listening doesn’t solve anything at all.
That kind of sounds like validation to me. 
Because sometimes it’s not a problem to be solved, but thoughts or feelings that the Haver just wants to share.
Because sometimes there is no solution.
Because sometimes the Haver needs to talk it through to think about it clearly.
Because sometimes the Haver will come to a solution by the end of the narrative without your help.
And again, offering a solution isn’t necessarily the problem here, in the OP. It’s the interrupting part that is the primary problem, for many reasons I and others have already stated.
I say hear it out and then offer a solution.
I had a g.f. who wasn’t the brightest of sparks, who would jump in, thinking that she understood the problem before three words were out of my mouth, with completely inane and irrelevant “solutions”.
If the Problem Haver starts to repeat themselves, or isn’t actually after a solution but wants to enjoy a bit of self pity, then all bets are off.
How many times on these boards have we had people ostensibly asking for for solutions, only for them rejecting each and every idea with "Oh I can’t do that because of such and such ", until everyone realises that they are there for the attention and enjoying a good wallow in the self pity mud hole.
Well, for me, if someone I care about is really unhappy about something, I want to know. And conversely, if I’m really unhappy about something, I’d hope that someone who cares about me would want to know. It makes me feel better to know that someone cares about how I’m feeling, and wants to fix my problem for me, even if they can’t.
And even if the problem can’t be fixed, perhaps it can at least be ameliorated in some way. If you had a horrible day at work, I can’t help that, but I can say, “Wow, that really sucks. I’m sorry you had to go through that. Leave the dishes, and let’s go get a beer.”
Heh, not if the answer is “no, you’re not right to be upset about this.”
Well yeah, there’s that.
I totally agree to shut up and listen. If this is an issue frequently, during a time when the partner is not venting, ask them what they would prefer. My husband and I have gotten into arguments before because he was venting to me about something and I kept coming in with solutions and getting irritated when he didn’t seem to want to hear them. And, of course, he would get annoyed because I was interrupting.
Finally, I asked him point-blank one day, “When you’re really mad, are you looking for someone to just listen or do you want me to help you?” His response was that he knew what he needed to do, but really needed someone to listen to him so he could blow off some steam.
In my relationship with my husband, our roles are usually flip-flopped. I’m generally the one jumping in with solutions; my husband is usually the one doing the venting.
And conversely, how many times have people prefaced a post with ‘I just want to vent,’ only to have posters jump in with solutions.