K-means implementations generally terminate once there's an iteration where the score doesn't improve. This happens when there is convergence to a local minimum or - less likely - the state hops between two nearby local minima with the same score. But it will terminate on something, and most of the time that something will be pretty good.

I saw your mention of Knuth elsewhere, I looked it up and he demanded that

> An algorithm must always terminate after a finite number of steps ... a very finite number, a reasonable number

This is a pretty niche characterization and almost certainly not what the original post was asking for. However, I concur that there is no guarantee on how quickly K-means terminates or on how good the output will be,. But... if you're going to be that strict about it you would even have to rule out the Simplex Algorithm, which everyone I've ever spoken to thinks of as an algorithm.