Do we have any idiom in English stating such a concept? "the problem got so bad/complex that it cannot be solved anymore"
You may use the idiomatic expression quagmire:
- a situation that is hard to deal with or get out of : a situation that is full of problems.
- That was six months ago, when the Defense secretary laughingly dismissed the idea that Iraq was, or could turn into, a quagmire.
- (informal) a situation in which no further progress can be made.
There are many colloquial phrases perhaps derived in spirit from this myth, for tasks which can't be completed, for example pushing water uphill with a rake and nailing jelly to a wall. These are quite common in UK engineering circles.
If it was impossible when it was given to you and your manager knew it but you didn't, the effect on your career might be serious. In this case the phrase poisoned chalice would be applicable. Strictly this lacks the sense of impossibility of fixing solving the problem, but a solution wouldn't be a success.
There is an interesting word for a problem so hard to solve within its (usually implied) rules but so important that someone breaks those rules in order to obtain a solution: a gordian knot problem, cutting the gordian knot.
I would say SNAFU, military slang for «Situation Normal: All Fucked Up». The problem/situation is horrible, but it's been so for long enough as to be accepted as the normal situation.
A project that has gotten so difficult as to be impossible is sometimes called a death march.
In project management, a death march is a project where the members feel it is destined to fail, or requires a stretch of unsustainable overwork. The general feel of the project reflects that of an actual death march because the members of the project are forced to continue the project by their superiors against their better judgment.
"Your project is doomed"
Doomed — TFD
marked for certain death
"the black spot told the old sailor he was doomed"
marked by or promising bad fortune
"their business venture was doomed from the start"
Forlorn hope — TFD
An undertaking that seems very unlikely to succeed.
"This plan you have is a forlorn hope and will never work out the way you want"
Lost cause — TFD
a futile attempt; a hopeless matter.
"Our campaign to have the new party on the ballot was a lost cause."
"Todd gave it up as a lost cause."
Losing battle — TFD
Well if you want to make sure nobody but computing scientists understand you, you could always say:
The problem is NP-complete.
But this actually means that you can devise a way (an algorithm) that would in theory solve the problem, but in practice it would take an infinite (or impractically long) time.
Is used to signify an insurmountable puzzle;
legend of Phrygian Gordium associated with Alexander the Great. It is often used as a metaphor for an intractable problem (disentangling an "impossible" knot) solved easily by loophole or "thinking outside the box" ("cutting the Gordian knot"):
One more option is:
The problem is incurable.
This metaphorical expression likens the problem to a disease that cannot be cured, and is especially suitable in situations where the problem has gotten worse over time or is too difficult to eliminate without causing harm or further problems, like some terminal illnesses.
Another option is:
The problem has spread like wildfire.
This clearly is applicable to problems that can spread quickly and cannot be easily doused. Not surprisingly, people who try to quell such problems are often described as trying to douse the issue.
There are several that connote having only bad choices available:
"Catch 22" (taken from the title of the book of the same name).
"between a rock and a hard place"
"damned if I do, damned if I don't."
"Tool-blocked" refers to a bolt or screw that is impossible to remove because the necessary tool won't fit or work in the space the fastener is in. It could be used metaphorically to mean a problem that you could try to solve but you would get in your own way.