It would depend on the problem.  There are algorithms that solve some problems that will always pick a locally optimal partial solution to yield a globally optimal solution at the end.  Algorithms such as Prim's Algorithm are like this.  I'm not sure if I'd call this intuition or anything related to that because it took intuition to understand that the algorithm could do this.  Intuition isn't as concrete as an algorithm in terms of concepts, so I'm not sure if we could capture that (at least yet).

Now if you were to extend this to any problem, I don't know and would highly doubt it.  Intuition is best suited for the mathematician, not the computer.  That's how we are able to devise new algorithms, and prove new theorems.

I don't think computers can use intuition as such, but there are ways to model what we humans call intuition, like ANN, CBR or fuzzy logic. I mean, intuition is usually based on a deep knowledge of the situation you are handling, up to the point that you can't actually put it in words, but you have a conviction on what to do (there is a very nice divulgation book on this kind of decisions called Blink). If you provide enough knowledge to a system, it can react at low level (without explicit models) to act one way or another (check purely reactive systems and behavior based systems). But a computer would not make a decision out of the blue. 

From my point of view, intuition contains the ability of decision making based on little information. Would it help to reduce the dimension of the feature space using the PCA for example and make some decision based on this limited amount of information? Does this correspond to the human way of thinking?

I think in the context of intuition it is important, how the locally optimal partial solution Daniel Page mentions, is generated.

At least chess-engines use heuristic evaluation functions, in which material and structures of positions are assessed. Probably there are algorithms that start to evaluate particular moves of particular figures first, depending on the actual position. The condition of the king is essential, so to speak. Generally, the best move of some move-options is chosen. But to come closer to intuition, it could be possible to merely evaluate move options, until a move which is good enough is found.

To go one step further, choose a (possible) move just from the (essential) features of the current position, without evaluating the resulting position. E.g. if there are still many figures on the board, capture the queen with a pawn if possible.

The question remains, how many features of a situation an algorithm must and may evaluate, so that it is "intuitive".

"And ye tell me, friends, that there is to be no dispute about taste and tasting? But all life is a dispute about taste and tasting!" - Friedrich Nietzsche

In my personal view there is only one main difference between "intuition" and "reasoning", namely, consciousness (of course, it is the next problem to define it..;-). Our intuitive decisions are based on the same knowledge as our logical decisions, but in the former case, we are not aware of how we come to a decision. Chess is a good example here, as a master chess player can often not explain, why a move is good, he just knows it is good. But again, this is a common phenomenon in our brain, as we move trained patterns more and more to unconsciousness. Playing chess for a chess master is like driving a car for an average person. It just came to my mind that following these lines, the decisions of a computer are in general intuitive, as it can usually not explain its decisions.

The intuition you are speaking about is more like "fastest recognition": in these terms, whereas a concrete context is given, the question is "how much things you should ascertain in order to have a positive response". Alas, your example may be troublesome (positively): that kind of recognition (love at first sight) seems to be based on the "whole" rather than parts. Undeniably, there are some elements that trigger the spark (i.e. the face, the voice or a behavior), but they act more like a defalcation, a "break" inside a cycle within the algorithm, which leads directly to a positive outcome instead of keeping it running. You can call these parts as "sufficient ground" for your recognition: once one is found, it turns the boolean control (checked or not) over all other traits on - after all, vices are discovered only in retrospect! 

Actually, problems rise when "nothing in particular says it it the right one, but it is": in this case, you have not sufficient features to manage a direct response - the conclusion is sealed like within a hologram - so you have to "ponder it all" without shortcuts; and there should be no hesitation: in order to attain the fastest recognition, the algorithm should have an overarching field of endeavor. 

In the paper, Four Models for  Decision Support System, Mirchandani and Pakath, Information and Management, 35(1), Jan 4, 1999, pp 31-42, the authors discuss a type of DSS called the Holsitic DSS. This discussion is relevant to your question. The discussion also draws upon the paper, 'Why Expert SYstems do not Exhibit Expertse," by H. Dreyfus and S. Dreyfus, IEEE Expert, 1(2), 1986, pp. 86-90. There, the authors distinguish between 5 stages of human expertise-- Novice, Advanced Beginner, Competent, Proficient., and Expert. The latter two stages, involve intuition guided problem recognition (Proficient and Expert) and intuition-guided action (Expert). The point of that paper was to argue that rule-based Expert systems  are hardly "expert" as it is incapable of intuition. Mirchandani and Pakath draw upon Dreyfus and Dreyfus' work and others to characterize Holistic systems. YOu may find one or both pieces useful. 

I think it all depends on what you mean by "intuitive"

1) Instinctive knowing (without the use of rational processes)

You might be able to reproduce the behaviour using probability theory. You don't directly model the process, just the likelihood of an event.

2) "An impression that something might be the case"

In my research area of "way finding" then yes there are algorithms for "intuitive wayfinding", where agents can navigate using visual cues/visual elements (i.e. signs or staircase) or heuristic behaviour tendencies such as first seeking out staircase/lifts (i.e. travelling between levels) and then along the horizontal to locate a location in a multilevel structures,